RD Sharma Class 12 Exercise 11.1 Higher Order Derivatives Solutions Maths

RD Sharma Class 12 Exercise 11.1 Higher Order Derivatives Solutions Maths - Download PDF Free Online

Edited By Kuldeep Maurya | Updated on Jan 27, 2022 02:55 PM IST

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RD Sharma Class 12 Solutions Chapter 11 Higher Order Derivatives - Other Exercise
Higher Order Derivatives Excercise:11.1
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Higher Order Derivatives Excercise:11.1

Higher Order Derivatives exercise 11.1 question 1(i)

Answer:
$6x+2sec^{2}x\: tan\, x$
Hint:
You must know about derivative of tan x
Given:
$x^{3}+tan\: x$
Solution:
$Let\: \: y=x^{3}+tan\: x$
$\begin{aligned} &\frac{d y}{d x}=3 x^{2}+\sec ^{2} x \quad \quad\left[\frac{d(\tan x)}{d x}=\sec ^{2} x \text { and } \frac{d x^{3}}{d x}=3 x^{2}\right] \\ &\frac{d^{2} y}{d x^{2}}=6 x+2 \sec x \cdot \sec x \tan x \frac{d \sec x}{d x}=\sec x \tan x \\ &\frac{d^{2} y}{d x^{2}}=6 x+2 \sec ^{2} x \tan x \end{aligned}$

Higher Order Derivatives exercise 11.1 question 1(ii)

Answer:
$\frac{-[\sin (\log x)+\cos (\log x)]}{x^{2}}$
Hint:
You must know about derivative of sin x & log x
Given:
$sin(log\: x)$
Solution:
$Let\: \: y=sin(log\: x)$
$\begin{aligned} &\left.\frac{d y}{d x}=\cos (\log x) \times \frac{d}{d x} \log x \quad \text { ( } \frac{d \sin x}{d x}=\cos x \right) \\ &\left.\frac{d y}{d x}=\cos (\log x) \times \frac{1}{x} \quad \text { ( } \frac{d \log x}{d x}=\frac{1}{x}\right) \\ &\frac{d y}{d x}=\frac{\cos (\log x)}{x} \end{aligned}$
Use quotient rule
$As\: \: \frac{u}{v}=\frac{u'v-v'u}{v^{2}}$
$W\! here\: \: v=x\: \, and \: \: u=cos(log\: x)$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{\left.x \frac{d}{d x}(\cos (\log x))-\cos (\log x)\right) \frac{d}{d x}(x)}{x^{2}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-x \sin (\log x) \cdot \frac{d(\log x)}{d x}-1 \cdot \cos (\log x)}{x^{2}}\left(\frac{d}{d x} x=1\right) \\ &\frac{d^{2} y}{d x^{2}}=\frac{-\sin (\log x) \cdot \frac{1}{x} x-\cos (\log x)}{x^{2}} \quad\left(\frac{d \log x}{d x}=\frac{1}{x}\right) \\ &\frac{d^{2} y}{d x^{2}}=\frac{-\sin (\log x)-\cos (\log x)}{x^{2}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-[\sin (\log x)+\cos (\log x)]}{x^{2}} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 1(iii)

Answer:
$-cosec^{2}x$
Hint:
You must know about derivative of sin x & log x
Given:
$log(sin\, x)$
Solution:
$Let\: \: y=log(sin\, x)$
$\begin{array}{ll} \frac{d y}{d x}=\frac{1}{\sin x} \frac{d \sin x}{d x} & \left(\frac{d \log x}{d x}=\frac{1}{x}\right) \\\\ \frac{d y}{d x}=\frac{1}{\sin x} \cos x & \left(\frac{d \sin x}{d x}=\cos x\right) \\\\ \frac{d y}{d x}=\cot x & \left(\frac{\cos x}{\sin x}=\cot x\right) \\\\ \frac{d^{2} y}{d x^{2}}=\cot x & \\\\ \frac{d^{2} y}{d x^{2}}=-cosec^{2} x & \left(\frac{d}{d x} \cot x=-cosec^{2} x\right) \end{array}$

Higher Order Derivatives exercise 11.1 question 1(iv)

Answer:
$-24e^{x}sin\: 5x+10e^{x}cos\: 5x$
Hint:
You must know about derivative of sin x & e^x
Given:
$e^{x}sin\: 5x$
Solution:
$Let\: \: y=e^{x}sin\: 5x$
Use multiplicative rule
$As\: \: UV=UV^{1}+U^{1}V$
Where U=e^x & V=sin 5x
$\begin{aligned} &\frac{d y}{d x}=e^{x} \frac{d}{d x} \sin 5 x+\frac{d}{d x} e^{x} \cdot \sin 5 x \\ &\frac{d y}{d x}=e^{x} \cdot \cos 5 x \frac{d}{d x} 5 x+e^{x} \cdot \sin 5 x \quad\left(\frac{d}{d x} e^{x}=e^{x}\right) \quad\left(\frac{d \sin 5 x}{d x}=\cos 5 x\right) \\ &\frac{d y}{d x}=e^{x} \cdot \cos 5 x \cdot 5+e^{x} \cdot \sin 5 x \quad \quad\left(\frac{d}{d x} 5 x=5\right) \\ &\frac{d y}{d x}=5 e^{x} \cos 5 x+e^{x} \sin 5 x \end{aligned}$
$\begin{aligned} &\frac{d}{d x}(\frac{d y}{d x})=\frac{d}{d x}(5 e^{x} \cos 5 x+e^{x} \sin 5 x) \end{aligned}$
Again use multiplication rule
$As\: \: UV=UV^{1}+U^{1}V$
Where U=e^x & V=sin 5x
U=e^x & V=cos 5x
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=5\left[e^{x} \frac{d}{d x} \cos 5 x+\frac{d}{d x} e^{x} \cdot \cos 5 x\right]+\left[e^{x} \frac{d}{d x} \sin 5 x+\frac{d}{d x} e^{x} \cdot \sin 5 x\right] \\ &\frac{d^{2} y}{d x^{2}}=5\left[e^{x} \cdot(-\sin 5 x) \frac{d}{d x} 5 x+e^{x} \cdot \cos 5 x\right]+\left[e^{x} \cos 5 x \frac{d}{d x} 5 x+e^{x} \cdot \sin 5 x\right] \\ &\left(\frac{d \sin x}{d x}=\cos x, \frac{d \cos x}{d x}=-\sin x, \frac{d 5 x}{d x}=5\right) \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=5\left[e^{x} \cdot(-\sin 5 x) 5+e^{x} \cdot \cos 5 x\right]+\left[e^{x} \cos 5 x .5+e^{x} \cdot \sin 5 x\right] \\ &\frac{d^{2} y}{d x^{2}}=\left[25 e^{x} \sin 5 x+5 e^{x} \cdot \cos 5 x\right]+\left[5 e^{x} \cos 5 x +e^{x} \cdot \sin 5 x\right] \\ &\frac{d^{2} y}{d x^{2}}=-24 e^{x} \sin 5 x+10 e^{x} \cdot \cos 5 x \end{aligned}$

Higher Order Derivatives exercise 11.1 question 1(v)

Answer:
$27e^{6x}cos3x-36e^{6x}sin3x$
Hint:
You must know about derivative of cos3x & e^6x
Given:
$e^{6x}cos3x$
Solution:
$Let\: \: y=e^{6x}cos3x$
Use multiplicative rule
$As\: \: UV=UV^{1}+U^{1}V$
Where U=e^6x and V=cos3x
$\begin{aligned} &\frac{d y}{d x}=e^{6 x} \frac{d}{d x} \cos 3 x+\frac{d}{d x} e^{6 x} \cdot \cos 3 x \\ &\frac{d y}{d x}=e^{6 x}-\sin 3 x \frac{d}{d x} 3 x+6 e^{6 x} \cos 3 x \quad\left(\frac{d \cos x}{d x}=-\sin x, \frac{d}{d x} e^{6 x}=e^{6 x} .6\right) \\ &\frac{d y}{d x}=-3 e^{6 x} \sin 3 x+6 e^{6 x} \cos 3 x \\ &\frac{d^{2} y}{d x^{2}}=-3 e^{6 x} \sin 3 x+6 e^{6 x} \cos 3 x \end{aligned}$
$As\: \: UV=UV^{1}+U^{1}V$
Where U=e^6x and V=sin3x
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=-3\left[e^{6 x} \cos 3 x(3)+e^{6 x} \cdot \sin 3 x .6\right]+6\left[e^{6 x} \cos 3 x \cdot 6+e^{6 x} \cdot(-\sin 3 x)\right] \\ &\left(\frac{d \sin 3 x}{d x}=3 \cos 3 x, \frac{d \cos 3 x}{d x}=3(-\sin 3 x), \frac{d e^{6 x}}{d x}=6 e^{6 x}\right) \\ &\frac{d^{2} y}{d x^{2}}=-3\left[3 e^{6 x} \cos 3 x+6 e^{6 x} \cdot \sin 3 x\right]+6\left[e^{6 x} \cos 3 x \cdot 6-3 e^{6 x} \cdot \sin 3 x\right] \\ &\frac{d^{2} y}{d x^{2}}=27 e^{6 x} \cos 3 x-36 e^{6 x} \sin 3 x \end{aligned}$

Higher Order Derivatives exercise 11.1 question 1(vi)

Answer:
$5x+6x\: log\: x$
HInt:
You must know about derivative of log x & x³
Given:
$x^{3}\: log\: x$
Solution:
$Let\: \: y=x^{3}\: log\: x$
Use multiplicative rule
As UV=UV¹+U¹V
Where U=x³ & V=log x
$\begin{aligned} &\frac{d y}{d x}=x^{3} \frac{d}{d x} \log x+\frac{d}{d x} x^{3} \log x \\ &\frac{d y}{d x}=x^{3} \frac{1}{x}+3 x^{2} \log x \quad\left(\frac{d}{d x} \log x=\frac{1}{x}, \frac{d}{d x} x^{3}=3 x^{2}\right) \\ &\ \end{aligned}$
Use multiplicative rule
As UV=UV¹+U¹V
Where U=x² & V=log x
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x} x^{2}+3\left[x^{2} \frac{d}{d x} \log x+\frac{d}{d x} x^{2} \cdot \log x\right] \\ &\left.\frac{d^{2} y}{d x^{2}}=2 x+3[x+2 x \cdot \log x] \quad \text { ( } \frac{d}{d x} x^{2}=2 x, \frac{d}{d x} \log x=\frac{1}{x}\right) \\ &\frac{d^{2} y}{d x^{2}}=2 x+3 x+6 x \cdot \log x \\ &\frac{d^{2} y}{d x^{2}}=5 x+6 x \log x \end{aligned}$

Higher Order Derivatives exercise 11.1 question 1(vii)

Answer:
$\frac{-2x}{(1+x^{2})^{2}}$
Hint:
You must know about derivative of tan^-1x
Given:
tan^-1x
Solution:
$Let\: \: y=tan^{-1}x$
$\begin{aligned} &\frac{d y}{d x}=\frac{1}{1+x^{2}} \quad\left(\frac{d \tan ^{-1} x}{d x}=\frac{1}{1+x^{2}}\right) \\ & \end{aligned}$
Use Quotient rule
$\begin{aligned} &\text { As } \frac{u}{v}=\frac{u^{1} v-v^{1} u}{v^{2}} \\ &\text { Where } v=1+x^{2} \& u=1 \\ &\frac{d^{2} y}{d x^{2}}=\frac{\left(1+x^{2}\right) \frac{d}{d x} 1-1 \cdot \frac{d}{d x}\left(1+x^{2}\right)}{\left(1+x^{2}\right)^{2}} \\ &\left.\frac{d^{2} y}{d x^{2}}=\frac{0\left(1+x^{2}\right)-1.2 x}{\left(1+x^{2}\right)^{2}} \quad \text { ( } \frac{d}{d x} 1=0, \frac{d}{d x}\left(1+x^{2}\right)=2 x\right) \\ &\frac{d^{2} y}{d x^{2}}=\frac{0-2 x}{\left(1+x^{2}\right)^{2}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-2 x}{\left(1+x^{2}\right)^{2}} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 1(viii) maths

Answer:
$-xcos\: x-2sin\: x$
Hint:
You must know about derivative of x cos x
Given:
x cos x
Solution:
$Let\: \: xcos\: x$
Use multiplicative rule
As UV=UV¹+U¹V
Where U=x & V=cos x
$\begin{aligned} &\frac{d y}{d x}=x \frac{d}{d x} \cos x+\frac{d}{d x} x \cos x \quad\left(\frac{d}{d x} x=1, \frac{d}{d x} \cos x=-\sin x\right) \\ &\frac{d y}{d x}=-x \sin x+\cos x \end{aligned}$
Use multiplicative rule
As UV=UV¹+U¹V
Where U=x & V=sin x
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=-\left(x \frac{d}{d x} \sin x+\frac{d}{d x} x \cdot \sin x\right)+\frac{d \cos x}{d x} \\ &\frac{d^{2} y}{d x^{2}}=-(x \cos x+\sin x)-\sin x \quad\left(\frac{d}{d x} \cos x=-\sin x\right) \\ &\frac{d^{2} y}{d x^{2}}=-x \cos x-\sin x-\sin x \\ &\frac{d^{2} y}{d x^{2}}=-x \cos x-2 \sin x \end{aligned}$

Higher Order Derivatives exercise 11.1 question 1(ix)

Answer:
$\frac{-(1+log\: x)}{x^{2}(log\: x)^{2}}$
Hint:
You must know about derivative of log(log x)
Given:
$log(log\: x)$
Solution:
$Let\: \:y= log(log\: x)$
$\begin{aligned} &\frac{d y}{d x}=\frac{d}{d x} \log (\log x) \quad\left(\frac{d \log x}{d x}=\frac{1}{x}\right) \\ &\frac{d y}{d x}=\frac{1}{\log x} \frac{d}{d x}(\log x) \\ &\frac{d y}{d x}=\frac{1}{\log x} \cdot \frac{1}{x} \\ &\frac{d y}{d x}=\frac{1}{x \log x} \end{aligned}$
Use Quotient rule
$\begin{aligned} &\text { As } \frac{u}{v}=\frac{u^{1} v-v^{1} u}{v^{2}}\\ &\text { Where } u=1 \& v=x \log x\\ &\text { Use multiplicative rule }\\ &\text { As } U V=U V^{1}+U^{1} V\\ &\text { Where } U=1 \& V=\log x \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{\frac{d}{d x} 1 x \log x-1 \frac{d}{d x} x \log x}{(x \log x)^{2}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{0 . x \log x-x \log x}{x^{2}(\log x)^{2}} \quad\left(\frac{d}{d x} \log x=\frac{1}{x}, \frac{d}{d x} 1=0, \frac{d}{d x} x=1\right) \\ &\frac{d^{2} y}{d x^{2}}=\frac{-\log x-1}{x^{2}(\log x)^{2}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-(1+\log x)}{x^{2}(\log x)^{2}} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 2

Answer:
$2e^{-x}sin\: x$
Hint:
You have to show
$\frac{d^{2}y}{dx^{2}}=2e^{-x}sin\: x$
Given:
If y=e^-xcos x, show that
$\frac{d^{2}y}{dx^{2}}=2e^{-x}sin\: x$
Solution:
Let y=e^-xcos x
Use multiplicative rule
$\begin{aligned}\text { As } U V=U V^{1}+U^{1} V \\ \text { Where } U=e^{-x} \& V=\cos x \\\frac{d y}{d x}=e^{-x} \frac{d}{d x} \cos x+\cos x \frac{d}{d x} e^{-x}\\ \frac{d y}{d x}=-e^{-x} \sin x-\cos x e^{-x} & \left(\frac{d \cos x}{d x}=-\sin x, \frac{d e^{-x}}{d x}=-1 e^{-x}\right) \end{aligned}$
Again differentiating w.r.t.x we get
Use multiplicative rule
$\begin{aligned} &\text { As } U V=U V^{1}+U^{1} V \\ &\text { Where } U=\cos x\: \&\: V=e^{-x} \\ &\frac{d^{2} y}{d x^{2}}=-\left[\sin x \cdot \frac{d}{d x} e^{-x}+e^{-x} \cdot \frac{d}{d x} \sin x+\cos x \frac{d}{d x} e^{-x}+e^{-x} \frac{d}{d x} \cos x\right] \\ &\frac{d^{2} y}{d x^{2}}=-\left[\sin x \cdot\left(-e^{-x}\right)+e^{-x} \cos x+\cos x\left(-e^{-x}\right)+e^{-x}(-\sin x)\right] \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=-\left[-e^{-x} \sin x+e^{-x} \cos x-e^{-x} \cos x-e^{-x} \sin x\right] \\ &\left(\frac{d}{d x} \sin x=\cos x, \frac{d}{d x} \cos x=-\sin x, \frac{d}{d x} e^{-x}=-1 e^{-x}\right) \\ &\frac{d^{2} y}{d x^{2}}=-\left[-2 \sin x e^{-x}\right] \\ &\frac{d^{2} y}{d x^{2}}=2 e^{-x} \sin x \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 3

Answer:
$\cos ^{2} x \frac{d^{2} y}{d x^{2}}-2 y+2 x=0$
Hint:
You have to show
$\cos ^{2} x \frac{d^{2} y}{d x^{2}}-2 y+2 x=0$
Given:
If y=x+tan x show that
$\cos ^{2} x \frac{d^{2} y}{d x^{2}}-2 y+2 x=0$
Solution:
Let y=x+tan x
$y=x+tan x\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (\frac{d\: tan\: x}{dx}=sec^{2}x,\: \frac{\mathrm{d} }{\mathrm{d} x}x=1)$
$\begin{aligned} &\frac{d y}{d x}=1+\sec ^{2} x \\ &\frac{d^{2} y}{d x^{2}}=2 \sec x(\sec x \tan x) \quad\left(\frac{d}{d x} \sec ^{2} x=2 \sec x(\sec x \tan x)\right. \\ &\frac{d^{2} y}{d x^{2}}=2 \sec ^{2} x \tan x \quad \quad\left(\sec ^{2} x=\frac{1}{\cos ^{2} x}, \tan x=\frac{\sin x}{\cos x}\right) \\ &\frac{d^{2} y}{d x^{2}}=\frac{2 \sin x}{\cos ^{3} x} \end{aligned}$
According to question
$\cos ^{2} x \frac{d^{2} y}{d x^{2}}-2 y+2 x=0$
substituting these values we get
$\begin{aligned} &\left(\frac{2 \sin x}{\cos x}\right)-2(x+\tan x)+2 x \\ &\frac{2 \sin x}{\cos x}-2 x-2 \tan x+2 x \\ &2 \tan x-2 x-2 \tan x+2 x=0 \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 4 maths

Answer:
$\frac{6}{x}$
HInt:
You must know about derivative of x³ & log x
Given:
If y=x³log x, Prove that
$\frac{d^{4}y}{dx^{4}}=\frac{6}{x}$
Solution:
Let y=x³log x
Use multiplicative rule
$As \: \: UV=UV^{1}+U^{1}V$
$\begin{aligned} &\text { Where } U=x^{3} \& V=\log x \\ &\frac{d y}{d x}=x^{3} \frac{d}{d x} \log x+\frac{d}{d x} x^{3} \log x \\ &\frac{d y}{d x}=x^{3} \frac{1}{x}+3 x^{2} \log x \quad\left(\frac{d \log x}{d x}=\frac{1}{x}, \frac{d}{d x} x^{3}=3 x^{2}\right) \\ &\frac{d y}{d x}=x^{2}+3 x^{2} \log x \end{aligned}$
Use multiplicative rule
$As \: \: UV=UV^{1}+U^{1}V$
$\begin{aligned} &\text { Where } U=x^{2} \& V=\log x \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=3\left(x^{2} \frac{d}{d x} \log x+\frac{d}{d x} x^{2} \log x\right)+\frac{d}{d x} x^{2} \\ &\frac{d^{2} y}{d x^{2}}=3\left(x^{2} \frac{1}{x}+2 x \log x\right)+2 x \quad\left(\frac{d \log x}{d x}=\frac{1}{x}, \frac{d}{d x} x^{2}=2 x\right) \\ &\frac{d^{2} y}{d x^{2}}=3 x+6 x \log x+2 x \\ &\frac{d^{2} y}{d x^{2}}=5 x+6 x \log x \end{aligned}$
Use multiplicative rule
$As \: \: UV=UV^{1}+U^{1}V$
$\begin{aligned} &\text { Where } U=x\: \&\: V=\log x \end{aligned}$
$\begin{aligned} &\frac{d^{3} y}{d x^{3}}=\frac{d}{d x} 5 x+6\left(x \frac{d}{d x} \log x+\frac{d}{d x} x \log x\right) \\ &\frac{d^{3} y}{d x^{3}}=5+6\left(x \frac{1}{x}+\log x\right) \quad\left(\frac{d \log x}{d x}=\frac{1}{x}, \frac{d}{d x} 5 x=5\right) \\ &\frac{d^{3} y}{d x^{3}}=5+6+6 \log x \end{aligned}$
$\begin{aligned} &\frac{d^{3} y}{d x^{3}}=11+6 \log x \\ &\left.\frac{d^{4} y}{d x^{4}}=0+\frac{6}{x} \quad \text { ( } \frac{d \log x}{d x}=\frac{1}{x}, \frac{d}{d x} 11=0\right) \\ &\frac{d^{4} y}{d x^{4}}=\frac{6}{x} \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 5

Answer:
$2cos\: x\: cosec^{3}x$
Hint:
You must know about how third derivatives be find
Given:
If y=log x(sin x). Prove that
$\frac{d^{3}y}{dx^{3}}=2cos\: x\: cosec^{3}x$
Solution:
$Let\: \: y=log (sin x)\; \; \; \; \; \; \; \; (\frac{d\: log\: x}{dx}=\frac{1}{x})$
$\begin{aligned} &\frac{d y}{d x}=\frac{1}{\sin x} \frac{d}{d x} \sin x \\ &\frac{d y}{d x}=\frac{1}{\sin x} \cos x \\ &\frac{d y}{d x}=\cot x \end{aligned}$
again differentiating we get
$\frac{d^{2} y}{d x^{2}}=-cosec^{2} x$ $(\frac{d\: cot\: x}{dx}=-cosec^{2}x)$
Again differentiating w.r.t. x we get
$\begin{aligned} &\frac{d^{3} y}{d x^{3}}=-2 \operatorname{cosec} x\cdot(-\cos ec x\cdot \cot x) \quad\left(\frac{d \operatorname{cosec}^{2} x}{d x}=-cosec\: x \cot x\right) \\ &\frac{d^{3} y}{d x^{3}}=-2 \cos x \operatorname{cosec}^{3} x \end{aligned}$

Higher Order Derivatives exercise 11.1 question 6

Answer:
$\frac{d^{2}y}{dx^{2}}+y=0$
Hint:
You have to know about how to find derivative of second order
Given:
If y=2sin x+3cos x,show that
$\frac{d^{2}y}{dx^{2}}+y=0$
Solution:
Let y=2sin x+3cos x
$\begin{aligned} &\frac{d y}{d x}=2 \cos x-3 \sin x \quad \quad\left(\frac{d \cos x}{d x}=-\sin x, \frac{d \sin x}{d x}=\cos x\right) \\ &\frac{d^{2} y}{d x^{2}}=-2 \sin x-3 \cos x \\ &\frac{d^{2} y}{d x^{2}}=-y \\ &\frac{d^{2} y}{d x^{2}}+y=0 \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 7

Answer:
$\frac{d^{2} y}{d x^{2}}=\frac{2 \log x-3}{x^{3}}$
Hint:
You have to know about derivative of
$\frac{ \log x}{x}$
Given:
$I\! f\: \:y= \frac{ \log x}{x}, show\: \: that$
$\frac{d^{2} y}{d x^{2}}=\frac{2 \log x-3}{x^{3}}$
Solution:
$Let \:y= \frac{ \log x}{x}$
Use quotient rule
$\begin{aligned} &\frac{u}{v}=\frac{u^{1} v-v^{1} u}{v^{2}} \\ &u=\log x\: \&\: v=x \\ &\frac{d y}{d x}=\frac{\log x}{x} \\ &\frac{d y}{d x}=\frac{x \frac{d}{d x} \log x-\log x \frac{d}{d x} x}{x^{2}} \quad\left(\frac{d \log x}{d x}=\frac{1}{x}\right) \\ &\frac{d y}{d x}=\frac{x \frac{1}{x}-\log x \cdot 1}{x^{2}} \\ &\frac{d y}{d x}=\frac{1-\log x}{x^{2}} \end{aligned}$
Use Quotient rule again
$\begin{aligned} &\frac{u}{v}=\frac{u^{1} v-v^{1} u}{v^{2}} \quad \\ &u=1-\log x \: \&\: v=x^{2} \\ &\frac{d^{2} y}{d x^{2}}=\frac{x^{2} \frac{d}{d x}(1-\log x)-\frac{d}{d x} x^{2}(1-\log x)}{\left(x^{2}\right)^{2}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{x^{2}\left(\frac{-1}{x}\right)-(1-\log x) 2 x}{x^{4}} \quad\left(\frac{d-\log x}{d x}=\frac{-1}{x}, \frac{d}{d x} x^{2}=2 x\right) \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{-x-2 x+2 x \log x}{x^{4}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{2 x \log x-3 x}{x^{4}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{2 x \log x-3}{x^{3}} \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 8 maths

Answer:
$\frac{-b^{4}}{a^{2}y^{3}}$
Hint:
You must know about derivative of
$tan\: \theta \: \: and\: \: sec\: \theta$
Given:
$I\! f\: \: x=a\: sec\: \theta ,y=b\: tan\: \theta$
Prove that
$\frac{d^{2}y}{dx^{2}}=\frac{-b^{4}}{a^{2}y^{3}}$
Solution:
$Let\: \: x=a\: sec\: \theta \\y=b\: tan\: \theta$
$\begin{aligned} &\frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right) \\ &x=a \sec \theta, y=b \tan \theta \end{aligned}$
$\begin{aligned} &\left.\frac{d x}{d \theta}=a \sec \theta \tan \theta \quad \text { ( } \frac{d \tan \theta}{d x}=\sec ^{2} x, \frac{d \sec \theta}{d x}=\sec \theta \tan \theta\right) \\ &\frac{d y}{d \theta}=b \sec ^{2} \theta \\ &\frac{d y}{d x}=\frac{b}{a} \frac{\sec \theta}{\tan \theta}=\frac{b}{a \sin \theta} \\ &\frac{d}{d \theta}\left(\frac{d y}{d x}\right)=-\frac{b}{a} \operatorname{cosec} \theta \cot \theta \end{aligned}$
$\begin{aligned} &=\frac{-\frac{b}{a} \operatorname{cosec} \theta \cot \theta}{a \sec \theta \tan \theta} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-b}{a^{2} \tan ^{3} \theta} y=b \tan \theta \\ &\frac{d^{2} y}{d x^{2}}=\frac{-b}{a^{2}\left(\frac{y}{b}\right)^{3} \theta} y=\frac{-b^{4}}{a^{2} y^{3}} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 9

Answer:
$\frac{sec^{3}\theta }{a\theta }$
Hint:
You must know about derivative of
$cos\: \theta \: \: and\: \: sin\: \theta$
Given:
$\begin{aligned} &\text { If } x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta) \text { Prove that }\\ &\frac{d^{2} x}{d \theta^{2}}=a(\cos \theta-\theta \sin \theta), \frac{d^{2} y}{d \theta^{2}}=a(\sin \theta-\theta \cos \theta), \frac{d^{2} y}{d x^{2}}=\frac{\sec ^{3} \theta}{a \theta} \end{aligned}$
Solution:
$\begin{aligned} &\text { Let } x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta) \end{aligned}$
Use multiplicative rule
$\begin{aligned} &\text { As } U V=U V^{1}+U^{1} V \\ &\text { Where } U=\theta \& V=\sin \theta \\ &\begin{array}{l} \frac{d x}{d \theta}=a[-\sin \theta+\theta \cos \theta+\sin \theta] \quad\left[\frac{d \sin \theta}{d x}=\cos \theta, \frac{d \cos \theta}{d x}=-\sin \theta\right] \\ \\ \frac{d x}{d \theta}=a \theta \cos \theta \end{array} \end{aligned}$
again
Use multiplicative rule
$\begin{aligned} &\text { As } U V=U V^{1}+U^{1} V \\ &\text { Where } U=\theta\: \&\: V=\cos \theta \\ &\frac{d y}{d \theta}=a[\cos \theta+\theta \sin \theta-\cos \theta] \\ &\frac{d y}{d \theta}=a \theta \sin \theta \end{aligned}$
Again Use multiplicative rule
$\begin{aligned} &\text { As } U V=U V^{1}+U^{1} V \\ &\text { Where } U=\theta \& V=\cos \theta \\ &\frac{d^{2} x}{d \theta^{2}}=a[-\theta \sin \theta+\cos \theta] \\ &\frac{d^{2} x}{d \theta^{2}}=a[\cos \theta-\theta \sin \theta] \end{aligned}$
Again Use multiplicative rule
$\begin{aligned} &\text { As } U V=U V^{1}+U^{1} V \\ &\text { Where } U=\theta \& V=\sin \theta \\ &\frac{d^{2} y}{d \theta^{2}}=a\{\theta \cos \theta+\sin \theta\} \\ &\frac{d^{2} x}{d \theta^{2}}=a(\cos \theta-\theta \sin \theta) \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d \theta^{2}}=a(\theta \cos \theta+\sin \theta) \\ &\frac{d y}{d x}=\frac{a \theta \sin \theta}{a \theta \cos \theta}=\tan \theta \\ &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)=\frac{d}{d \theta}\left(\frac{d y}{d x}\right) \frac{d \theta}{d x} \end{aligned}$
$\begin{aligned} &\frac{d}{d \theta}(\tan \theta) \times \frac{1}{a \theta \cos \theta} \\ &\frac{\sec ^{2} \theta}{a \theta \cos \theta} \\ &\frac{d^{2} y}{d x^{2}}=\frac{\sec ^{3} \theta}{a \theta} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 10

Answer:
$2e^{x}cos(x+\frac{\pi }{2})$
Hint:
You must know about derivative of e^x cos x
Given:
If y = e^x cos x, prove that
$\frac{d^{2}y}{dx^{2}}=2e^{x}cos(x+\frac{\pi }{2})$
Solution:
Let y = e^x cos x
Use multiplicative rule
$\begin{aligned} &\text { As } U V=U V^{1}+U^{1} V \\ &\text { Where } U=e^{x}\: \&\: V=\cos x \\ &\frac{d y}{d x}=e^{x}(-\sin x)+e^{x} \cos x \quad \quad\left(\frac{d}{d x} \cos x=\sin x\right) \end{aligned}$
Again Use multiplicative rule
Differentiating again
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=-\left[e^{x} \cos x+\sin x e^{x}\right]+\left[-e^{x} \sin x+e^{x} \cos x\right] \\ &\frac{d^{2} y}{d x^{2}}=-2 \sin x e^{x} \\ &\frac{d^{2} y}{d x^{2}}=2 e^{x} \cos \left(x+\frac{\pi}{2}\right) \end{aligned}$

Higher Order Derivatives exercise 11.1 question 11

Answer:
$\frac{-b}{a^{2}y^{3}}$
Hint:
You must know about derivative of second order
Given:
$\begin{aligned} &{\text { If } x}=a \cos \theta, y=b \sin \theta \\ &\text { Show that } \frac{d^{2} y}{d x^{2}}=\frac{-b}{a^{2} y^{3}} \end{aligned}$
Solution:
$\begin{aligned} &{\text {Let}\: \: x}=a \cos \theta, y=b \sin \theta \\ \end{aligned}$
$\begin{aligned} &\frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{\frac{d}{d \theta}\left(\frac{d y}{d x}\right)}{\frac{d x}{d \theta}} \\ &x=a \cos \theta, y=b \sin \theta \\ &\frac{d x}{d \theta}=-a \sin \theta \frac{d y}{d \theta}=b \cos \theta \\ &\frac{d y}{d x}=\frac{-b}{a} \cot \theta \end{aligned}$
$\begin{aligned} &\frac{\frac{\mathrm{d} }{\mathrm{d} \theta }(\frac{dy}{dx})}{\frac{dx} {d\theta }}=\frac{\frac{b}{a} \cos e c^{2} \theta}{-a \sin \theta} \quad\left(\frac{d}{d \theta} \cot \theta=\cos e c^{2} \theta\right) \\ &\frac{-b \times b^{3}}{a^{2} \sin ^{3} \theta \times b^{3}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-b}{a^{2} y^{3}} \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 12 maths

Answer:
$\frac{32}{27a}$
Hint:
You must know about derivative of
$\frac{cos^{3}\theta}{sin^{3}\theta }$
Given:
$\begin{aligned} &{\text { If } x=a\left(1-\cos ^{3} \theta\right), y=a \sin ^{3} \theta} \\ &\text { Prove that } \frac{d^{2} y}{d x^{2}}=\frac{32}{27 a} \; a t\; \theta=\frac{\pi}{6} \end{aligned}$
Solution:
$\begin{aligned} &{\text { Let } x=a\left(1-\cos ^{3} \theta\right), y=a \sin ^{3} \theta} \end{aligned}$
$\begin{aligned} &\frac{d y}{d \theta}=3 a \sin ^{2} \theta \cos \theta\left(\frac{d}{d \theta} \sin ^{3} \theta=3 a \sin ^{2} \theta \cos \theta\right) \\ &\frac{d y}{d \theta}=3 \cos ^{2} \theta \sin \theta \ \left(\frac{d}{d \theta} \cos ^{3} \theta=3 \cos ^{2} \theta \sin \theta\right) \\ &\frac{d y}{d x}=\tan \theta \quad \quad\left(\frac{d}{d \theta} \tan \theta=\sec ^{2} \theta\right) \end{aligned}$
$\begin{aligned} &\frac{\frac{d}{d \theta}\left(\frac{d y}{d x}\right)}{\frac{d x}{d \theta}}=\frac{\sec ^{2} \theta}{3 a \cos ^{2} \theta \sin \theta}\\ &\frac{d^{2} y}{d x^{2}}=\frac{\sec ^{4} \theta}{3 a \sin \theta} \quad\left(\theta=\frac{\pi}{6}\right)\\ &\frac{d^{2} y}{d x^{2}}=\frac{\sec ^{4}\left(\frac{\pi}{6}\right)}{3 a \sin \left(\frac{\pi}{6}\right)}=\frac{32}{27 a} \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 13

Answer:
$\frac{-a}{y^{2}}$
HInt:
You must know about derivative of second order
Given:
$\begin{aligned} &{\text { If } x=a(\theta+\sin \theta), y=a(1+\cos \theta)} \\ &\text { Prove that } \frac{d^{2} y}{d x^{2}}=\frac{-a}{y^{2}} \end{aligned}$
Solution:
$\begin{aligned} &{\text { Let } x=a(\theta+\sin \theta), y=a(1+\cos \theta)} \\ \end{aligned}$
$\begin{aligned} &\left.\frac{d x}{d \theta}=a[1+\cos \theta] \quad \text { ( } \frac{d}{d \theta} \theta=1, \frac{d}{d \theta} \sin \theta=\cos \theta\right) \\ &\frac{d y}{d \theta}=a(-\sin \theta) \\ &\frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{-\sin \theta}{(1+\cos \theta)} \end{aligned}$
$\begin{aligned} &\sin 2 \theta=2 \sin \theta \cos \theta \\ &\cos 2 \theta=2 \cos ^{2} \theta-1 \\ &\frac{d y}{d x}=\frac{-2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^{2} \frac{\theta}{2}} \end{aligned}$
$\begin{aligned} &\frac{d y}{d x}=-\tan \frac{\theta}{2} \\ &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)=\frac{d}{d \theta}\left(\frac{d y}{d x}\right) \frac{d \theta}{d x} \\ &\frac{d}{d \theta}\left(-\tan \frac{\theta}{2}\right) \cdot \frac{1}{a(1+\cos \theta)} \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{-1}{2} \sec ^{2} \frac{\theta}{2} \cdot \frac{1}{a(1+\cos \theta)} \\ &=\frac{-\sec ^{2} \frac{\theta}{2}}{2 a(1+\cos \theta)} \\ &\frac{-a}{y^{2}}=\frac{-a}{a(1+\cos \theta)^{2}}=\frac{-a}{a^{2}(1+\cos \theta)(1+\cos \theta)} \end{aligned}$
$\begin{aligned} &\frac{-1}{a\left(1+2 \cos ^{2} \frac{\theta}{2}-1\right)(1-\cos \theta)} \\ &=\frac{-1}{2 a \cos ^{2} \frac{\theta}{2}(1+\cos \theta)}=\frac{-\sec ^{2} \frac{\theta}{2}}{2 a(1+\cos \theta)} \end{aligned}$
$\begin{aligned} &R H L:-\frac{-a}{y^{2}}=\frac{-a}{a(1+\cos \theta)^{2}}=\frac{-1}{a\left(1+2 \cos ^{2} \frac{\theta}{2}-1\right)(1-\cos \theta)} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-\sec ^{2} \frac{\theta}{2}}{2 a(1+\cos \theta)}=\frac{-a}{y^{2}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-a}{y^{2}} \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 14

Answer:
$\frac{-cosec\: x\frac{4\theta }{2}}{4a}$
Hint:
You must know about derivative of
$cos\: \theta \: and\: sin\: \theta$
Given:
$\begin{aligned} &\text { If } x=a(\theta-\sin \theta), y=a(1+\cos \theta) \\ &\text { Find } \frac{d^{2} y}{d x^{2}} \end{aligned}$
Solution:
$\begin{aligned} &\text { Let } x=a(\theta-\sin \theta), y=a(1+\cos \theta) \end{aligned}$
$\begin{aligned} &\frac{d x}{d \theta}=a[1-\cos \theta] \; \; \; \; \; ......(1)\\ &\frac{d y}{d \theta}=a(-\sin \theta)\; \; \; \; \; ......(2) \\ &\text { Now } \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{\sin \theta}{(1-\cos \theta)} \end{aligned}$
Using identity
$\begin{aligned} &\sin 2 \theta=2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \\ &2 \sin ^{2} \frac{\theta}{2}=1-\cos \theta \\ &\cos 2 \theta=1-2 \sin ^{2} \theta \\ &2 \sin ^{2} \theta=1-\cos 2 \theta \end{aligned}$
$\begin{aligned} &\frac{d y}{d x}=\frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \sin ^{2} \frac{\theta}{2}} \\ &\frac{d y}{d x}=\cot \frac{\theta}{2} \end{aligned}$
Now diff on both sides
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)=\frac{d}{d \theta}\left(\frac{d y}{d x}\right) \frac{d \theta}{d x} \\ &\frac{d^{2} y}{d x^{2}}=-\operatorname{cosec}^{2} \theta \cdot \frac{1}{2} \bullet \frac{1}{a(1-\cos \theta)} \\ &\frac{d^{2} y}{d x^{2}}=\operatorname{cosec}^{2} \frac{\theta}{2} \bullet \frac{1}{2} \bullet \frac{1}{a 2 \sin ^{2} \frac{\theta}{2}} \end{aligned}$
$\begin{aligned} &=\frac{-\cos e c^{2} \frac{\theta}{2} \times \cos e c^{2} \frac{\theta}{2}}{4 a} \\&\frac{d^{2} y}{d x^{2}}=\frac{\cos e c^{4} \frac{\theta}{2}}{4 a} \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 15

Answer:
$\frac{-1}{a}$
Hint:
$\begin{aligned} &{{\text { }}\text { You must know about derivative of } \cos \theta\: \&\: \sin \theta}\\ \end{aligned}$
Given:
$\begin{aligned} &\text { If } x=a(1-\cos \theta), y=a(\theta+\sin \theta)\\ &\text { Prove that } \frac{d^{2} y}{d x^{2}}=\frac{-1}{a} \: and\: \theta=\frac{\pi}{4}\\ \end{aligned}$
Solution:
$\begin{aligned} &{{\text { Let }} x=a(1-\cos \theta), y=a(\theta+\sin \theta)} \end{aligned}$
$\begin{aligned} &\frac{d x}{d \theta}=a \sin \theta \\ &\frac{d y}{d \theta}=a(1+\cos \theta) \\ &\frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{1+\cos \theta}{\sin \theta} \end{aligned}$
Use Quotient rule
$\begin{aligned} &\text { As } \frac{u}{v}=\frac{u^{1} v-v^{1} u}{v^{2}} \\ &\text { Where } u=1+\cos \theta \: \&\: v=\sin \theta \\ &\frac{d}{d \theta}\left(\frac{d y}{d x}\right)=\frac{\sin \theta(-\sin \theta)-(1+\cos \theta) \cos \theta}{\sin ^{2} \theta} \end{aligned}$
$\begin{aligned} &\frac{d}{d \theta}\left(\frac{d y}{d x}\right)=\frac{-1+\cos \theta}{(1-\cos \theta)(1+\cos \theta)} \\ &\frac{d^{2} y}{d x^{2}}=\frac{\frac{-1}{1-\cos \theta}}{a \sin ^{2} \theta} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-1}{a \sin \theta(1-\cos \theta)} \quad\left(\theta=\frac{\pi}{2}\right) \end{aligned}$
$\begin{aligned} \frac{d^{2} y}{d x^{2}} &=\frac{-1}{a \sin \frac{\pi}{2}\left(1-\cos \frac{\pi}{2}\right)} \\ \frac{d^{2} y}{d x^{2}} &=\frac{-1}{a} \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 16 maths

Answer:
$\frac{-1}{a}$
Hint:
$\begin{aligned} &{{\text { }}\text { You must know about derivative of } \cos \theta\: \&\: \sin \theta}\\ \end{aligned}$
Given:
$\begin{aligned} &\text { If } x=a(1+\cos \theta), y=a(\theta+\sin \theta)\\ &\text { Prove that } \frac{d^{2} y}{d x^{2}}=\frac{-1}{a}\: and\: \theta=\frac{\pi}{4}\\ \end{aligned}$
Solution:
$\begin{aligned} &{{\text { Let }} x=a(1+\cos \theta), y=a(\theta+\sin \theta)} \end{aligned}$
$\begin{aligned} &\frac{d x}{d \theta}=-a \sin \theta \\ &\frac{d y}{d \theta}=a(1+\cos \theta) \\ &\frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{1+\cos \theta}{-\sin \theta} \end{aligned}$
Use Quotient rule
$\begin{aligned} &\text { As } \frac{u}{v}=\frac{u^{1} v-v^{1} u}{v^{2}} \\ &\text { Where } u=1+\cos \theta \& v=\sin \theta \\ &\frac{d}{d \theta}\left(\frac{d y}{d x}\right)=\frac{-\sin \theta(-\sin \theta)+(1+\cos \theta) \cos \theta}{\sin ^{2} \theta} \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{1+\cos \theta}{\sin ^{2} \theta} \cdot \frac{-1}{a \sin \theta} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-(1+\cos \theta)}{a \sin ^{2} \theta} \\ &\left.\frac{d^{2} y}{d x^{2}}=\frac{-\left(1+\cos \frac{\pi}{2}\right)}{a \sin ^{2} \frac{\pi}{2}} \quad (\theta =\frac{\pi}{2}\right) \\ &\frac{d^{2} y}{d x^{2}}=\frac{-1}{a} \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 17

Answer:
$\begin{aligned} &\text { } 3 \sin ^{2} \theta\left[5 \cos ^{2} \theta-1\right]\\ \end{aligned}$
Hint:
$\begin{aligned} &\text { You must know about derivative of } \cos \theta\: \&\: \sin \theta\\ \end{aligned}$
Given:
$\begin{aligned} &\text { If } x=\cos \theta, y=\sin ^{3} \theta\\ &\text { Prove that } y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=3 \sin ^{2} \theta\left[5 \cos ^{2} \theta-1\right] \end{aligned}$
Solution:
$\begin{aligned} &\text { Let } x=\cos \theta, y=\sin ^{3} \theta\\ \end{aligned}$
$\begin{aligned} &\frac{d x}{d \theta}=-\sin \theta \\ &\frac{d y}{d \theta}=3 \sin ^{2} \theta \cos \theta \\ &\frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{3 \sin ^{2} \theta \cos \theta}{-\sin \theta}=3 \sin \theta \cos \theta \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=d\left(\frac{\frac{d y}{d x}}{d x}\right)=\frac{d\left(\frac{d y}{d \theta}\right) d \theta}{\frac{d x}{d \theta}} \\ &=\frac{-\cos ^{2} \theta \cdot 3+3 \sin ^{2} \theta}{-\sin \theta} \end{aligned}$
$\begin{aligned} &y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=\frac{\sin ^{3} \theta\left(3 \cos ^{2} \theta-3 \sin ^{2} \theta\right)}{\sin \theta} \\ &y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=3 \sin ^{2} \theta\left[5 \cos ^{2} \theta-1\right] \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 18

Answer:
0
Hint:
You must know about derivative of
$cos\: \theta\: and\: tan\: \theta$
Given:
$I\! f\: y=sin(sin\: x)$
$\begin{aligned} &Prove\: \: that\: \: \frac{d^{2} y}{d x^{2}}+\tan x \frac{d y}{d x}+y \cos ^{2} x =0\\ \end{aligned}$
Solution:
$Let\: y=sin(sin\: x)$
$\begin{aligned} &\frac{d y}{d x}=\cos (\sin x) \cdot \cos x \\ &\frac{d^{2} y}{d x^{2}}=-\cos (\sin x)(\sin x)+\cos x(-\sin (\sin x) \cos x) \\ &\frac{d^{2} y}{d x^{2}}=-\sin x \cos (\sin x)-\cos ^{2} x(\sin (\sin x)) \end{aligned}$
$\begin{aligned} &L H S:-\frac{d^{2} y}{d x^{2}}+\tan x \frac{d y}{d x}+y \cos ^{2} x \\ &-\sin x \cos (\sin x)-\cos ^{2} x(\sin (\sin x))+\tan x \cos x \cos (\sin x)+\sin (\sin x) \cos ^{2} x \\ &-\sin x(\cos (\sin x))+\frac{\sin x}{\cos x} \cdot \cos x \cdot \cos (\sin x) \\ &-\sin x \cos x \sin x+\sin x \cos x \sin x \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 19

Answer:
0
HInt:
You must know about derivative of sin pt
Given:
$I\! f\; x=sin\; t,\; y=sin\; pt$
$\text { Prove that }\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+p^{2} y=0$
Solution:
$Let\; x=sin\; t,\; y=sin\; pt$
$\begin{aligned} &\frac{d x}{d t}=\cos t \text { and } \frac{d y}{d t}=p \cos p t \\ &\text { Now } \frac{d y}{d x}=\frac{p \cos p t}{\cos t} \\ &\frac{d y}{d x} \cos t=p \cos p t \end{aligned}$
$\begin{aligned} &\left(\frac{d y}{d x}\right)^{2}\left(1-\sin ^{2} t\right)=p^{2}\left(1-\sin ^{2} p t\right) \\ &\left(\frac{d y}{d x}\right)^{2}\left(1-x^{2}\right)=p^{2}\left(1-y^{2}\right) \end{aligned}$
Differentiating with x
$\begin{aligned} &\left(1-x^{2}\right) 2 \frac{d y}{d x} \times \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}(-2 x) \\ &=-p^{2} \times 2 y \frac{d y}{d x} \\ &2 \frac{d y}{d x}\left[\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}\right]=p^{2} y 2 \frac{d y}{d x} \\ &\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+p^{2} y=0 \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 20 maths

Answer:
0
Hint:
You must know about derivative of sin^-1 x
Given:
$I\! f y=(sin^{-1}x)^{2} \; \; Prove\: that\: \: (1-x^{2})y_{2}-xy_{1}-2=0$
Solution:
$Let\; \; y=(sin^{-1}x)^{2}$
$\begin{aligned} &\frac{d y}{d x}=2 \sin ^{-1} x\left(\frac{1}{\sqrt{1-x^{2}}}\right) \quad\left(\frac{d}{d x} \sin ^{-1} x=\left(\frac{1}{\sqrt{1-x^{2}}}\right)\right) \\ &\frac{d^{2} y}{d x^{2}}=\frac{2 \sqrt{1-x^{2}}\left(\frac{1}{\sqrt{1-x^{2}}}\right)-\sin ^{-1} x \times \frac{1}{2}\left(\frac{-2 x}{\sqrt{1-x^{2}}}\right)}{1-x^{2}} \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{2\left(1+\frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}}\right)}{1-x^{2}} \\ &\left(1-x^{2}\right) y_{2}=\frac{2+2 x \sin ^{-1} x}{\sqrt{1-x^{2}}} \\ &\left(1-x^{2}\right) y_{2}-y_{1} x-2=0 \end{aligned}$
Hence proved

Higher Order Derivatives exercise 11.1 question 21

Answer:
Proved
Hint:
You must know about the derivative of exponential function and tangent inverse $x$
Given:
$y=e^{tan^{-1}x},\; \; Prove\: (1+x^{2})y_{2}+(2x-1)y_{1}=0$
Solution:
$Let\; \; y=e^{tan^{-1}x}$
$\begin{aligned} &\frac{d y}{d x}=e^{\tan ^{-1} x} \times \frac{1}{\left(1+x^{2}\right)} \\ &\left(x^{2}+1\right) \frac{d y}{d x}=e^{\operatorname{tan}^{-1} x} \\ &\left(x^{2}+1\right) \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=e^{\tan ^{-1} x} \times \frac{1}{\left(1+x^{2}\right)} \end{aligned}$
$\begin{aligned} &\left(x^{2}+1\right) \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=\frac{d y}{d x}\\ &\left(x^{2}+1\right) \frac{d^{2} y}{d x^{2}}+(2 x-1) \frac{d y}{d x}=0\\ &\text { or }\\ &\left(x^{2}+1\right) y_{2}+(2 x-1) y_{1}=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 22

Answer:
Proved
Hint:
You must know about the derivative of logarithm function and cos x and sin x
Given:
$y=3cos(log\: x)+4sin(log\: x)$
Solution:
$Let\: \: y=3cos(log\: x)+4sin(log\: x)$
$\begin{aligned} &\text { Differentiating both sides w.r.t } x\\ &\frac{d y}{d x}=-3 \sin (\log x) \frac{d(\log x)}{d x}+4 \cos (\log x) \frac{d(\log x)}{d x} \end{aligned}$
$\begin{aligned} &\text { By using product rule of derivation }\\ &\frac{d y}{d x}=\frac{-3 \sin (\log x)}{x}+\frac{4 \cos (\log x)}{x}\\ &x \frac{d y}{d x}=-3 \sin (\log x)+4 \cos (\log x) \end{aligned}$
$\begin{aligned} &\text { Again differentiating both sides w.r.t } x \text { , }\\ &x \frac{d}{d x}\left(\frac{d y}{d x}\right)+\frac{d y}{d x} \frac{d}{d x}(x)=\frac{d}{d x}[-3 \sin (\log x)+4 \cos (\log x)] \end{aligned}$
$\begin{aligned} &\text { By using product rule of derivation, }\\ &x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}(1)=-3 \cos (\log x) \frac{d}{d x}(\log x)-4 \sin (\log x) \frac{d}{d x}(\log x)\\ &x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}(1)=\frac{-3 \cos (\log x)}{x}-\frac{4 \sin (\log x)}{x} \end{aligned}$
$\begin{aligned} &x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=\frac{-[3 \cos (\log x)+4 \sin (\log x)]}{x} \\ &x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=-[3 \cos (\log x)+4 \sin (\log x)] \\ &x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=-y \end{aligned}$
$\begin{aligned} &\therefore x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0\\ &\text { or }\\ &x^{2} y_{2}+x y_{1}+y=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 23

Answer:
Proved
Hint:
You must know about the derivative of exponential function
Given:
$y=e^{2x}(ax+b), \: \: show\: \: that\: \: y_{2}-4y_{1}+4y=0$
Solution:
$y=e^{2x}(ax+b)\; \; \; \; \; \; \; \; \; ......(1)$
$\begin{aligned} &\text { By using product rule of derivation }\\ &\frac{d y}{d x}=e^{2 x} \frac{d y}{d x}(a x+b)+(a x+b) \frac{d}{d x} e^{2 x} \\ &\frac{d y}{d x}=a e^{2 x}+2(a x+b) e^{2 x}\\ &\frac{d y}{d x}=e^{2 x}(a+2 a x+2 b) \end{aligned}$ $.....(2)$
$\begin{aligned} &\text { Again differentiating both sides w.r.t } x, \text { using product rule }\\ &\frac{d^{2} y}{d x^{2}}=e^{2 x} \frac{d y}{d x}(a+2 a x+2 b)+(a+2 a x+2 b) \frac{d}{d x} e^{2 x}\\ &\frac{d^{2} y}{d x^{2}}=2 e^{2 x}+2(a+2 a x+2 b) e^{2 x} \quad \ldots .(3) \end{aligned}$
$In\; order\; to\; prove\; the\; expression\; try\; to\; get\; the\; required\; f\! orm \\Subtracting\; 4 \times equation\; (2)\; f\! rom\; equation (3)\\ \frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}=2 e^{2 x}+2(a+2 a x+2 b) e^{2 x}-4 e^{2 x}(a+2 a x+2 b)\\ \frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}=2 a e^{2 x}-2 e^{2 x}(a+2 a x+2 b)\\ \frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}=-4 e^{2 x}(a x+b)$
$\begin{aligned} &\text { Using equation }(1)\\ &\frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}=-4 y\\ &\frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}+4 y=0\\ &y_{2}-4 y_{1}+4 y=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 24 maths

Answer:
Proved
Hint:
You must know about the derivative of sin function and logarithm function
Given:
$x=\sin \left(\frac{1}{a} \log y\right) \text { show that }\left(1-x^{2}\right) y_{2}-x y_{1}-a^{2} y=0$
Solution:
$\begin{aligned} &x=\sin \left(\frac{1}{a} \log y\right)\\ &\log y=a \sin ^{-1} x\\ &y=e^{a \sin ^{-1} x}\; \; \; \; \; \; \; .....(1) \end{aligned}$
$\begin{aligned} &\text { Let } t=a \sin ^{-1} x\\ &\frac{d t}{d x}=\frac{a}{\sqrt{1-x^{2}}} \quad\left[\frac{d}{d x} \sin ^{-1} x=\frac{1}{\sqrt{1-x^{2}}}\right]\\ &\text { and } y=e^{t}\\ &\frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}\\ &\frac{d y}{d x}=e^{t} \frac{a}{\sqrt{1-x^{2}}}=\frac{a e^{a \sin ^{-1} x}}{\sqrt{1-x^{2}}}\; \; \; \; \; \; \; \; .......(2) \end{aligned}$
$\begin{aligned} &\text { Again differentiating both sides }\\ &\frac{d^{2} y}{d x^{2}}=a e^{a \sin ^{-1} x} \frac{d}{d x}\left(\frac{1}{\sqrt{1-x^{2}}}\right)+\frac{a}{\sqrt{1-x^{2}}} \frac{d}{d x} e^{a \sin ^{-1} x} \end{aligned}$
$\begin{aligned} &\text { Using chain rule and equation }\\ &\frac{d^{2} y}{d x^{2}}=\frac{-a e^{a \sin ^{-1} x}}{2\left(1-x^{2}\right) \sqrt{1-x^{2}}}(-2 x)+\frac{a^{2} e^{a \sin ^{-1} x}}{\left(1-x^{2}\right)}\\ &\frac{d^{2} y}{d x^{2}}=\frac{x a e^{a \sin ^{-1} x}}{\left(1-x^{2}\right) \sqrt{1-x^{2}}}+\frac{a^{2} e^{a \sin ^{-1} x}}{\left(1-x^{2}\right)}\\ &\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}=a^{2} e^{a \sin ^{-1} x}+\frac{x a e^{a \sin ^{-1} x}}{\sqrt{1-x^{2}}} \end{aligned}$
$\begin{aligned} &\text { Using equation }(1) \text { and }(2)\\ &\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}=a^{2} y+x \frac{d y}{d x}\\ &\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-a^{2} y-x \frac{d y}{d x}=0\\ &\left(1-x^{2}\right) y_{2}-x y_{1}-a^{2} y=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 25

Answer:
Proved
HInt:
You must know the derivative of logarithm and tangent inverse $x$
Given:
$\log y=\tan ^{-1} x, \text { show }\left(1+x^{2}\right) y_{2}+(2 x-1) y_{1}=0$
Solution:
$\begin{aligned} &\log y=\tan ^{-1} x\\ &\text { Differentiate the equation w.r.t } x\\ &\frac{1}{y} \frac{d y}{d x}=\frac{1}{1+x^{2}}\\ &1+x^{2} \frac{d y}{d x}=y \end{aligned}$
$Di\! f\! ferentiate\; again\\ \left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}(2 x)=\frac{d y}{d x}\\ \left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+(2 x-1) \frac{d y}{d x}=0\\ or\\ \left(1+x^{2}\right) y_{2}+(2 x-1) y_{1}=0$

Higher Order Derivatives exercise 11.1 question 26

Answer:
Proved
Hint:
You must know the derivative of logarithm and tangent inverse $x$
Given:
$y=\tan ^{-1} x, \text { show }\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+(2 x) \frac{d y}{d x}=\frac{d y}{d x}$
Solution:
$y=\tan ^{-1} x$
$Di\! f\! f\! erentiate\; the\; equation\; w.r.t \; x \\ \frac{d y}{d x}=\frac{1}{1+x^{2}}\\ \left(1+x^{2}\right) \frac{d y}{d x}=1 \\ Di\! f\! f\! erentiate\; again \\ \left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+(2 x) \frac{d y}{d x}=0 \\ or \\ \left(1+x^{2}\right) y_{2}+(2 x) y_{1}=0$

Higher Order Derivatives exercise 11.1 question 27

Answer:
Proved
Hint:
You must know the derivative of logarithm and tangent inverse $x$
Given:
$y=\left\{\log \left(x+\sqrt{x^{2}+1}\right)\right\}^{2}, \text { show }\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=2$
Solution:
$y=\left\{\log \left(x+\sqrt{x^{2}+1}\right)\right\}^{2}$
$\begin{aligned} &\text { Differentiate the equation w.r.t } x\\ &\frac{d y}{d x}=2 \log \left(x+\sqrt{x^{2}+1}\right) \cdot \frac{1}{x+\sqrt{x^{2}+1}} \times\left(1+\frac{2 x}{2 \sqrt{x^{2}+1}}\right)\\ &=\frac{2 \log \left(x+\sqrt{x^{2}+1}\right)}{x+\sqrt{x^{2}+1}} \times \frac{x+\sqrt{x^{2}+1}}{\sqrt{x^{2}+1}}\\ &=\sqrt{x^{2}+1} \cdot \frac{d y}{d x}=2 \log \left(x+\sqrt{x^{2}+1}\right) \end{aligned}$
$\begin{aligned} &\text { Differentiate again }\\ &\frac{d^{2} y}{d x^{2}} \sqrt{1+x^{2}}+\frac{2 x}{2 \sqrt{1+x^{2}}} \frac{d y}{d x}=\frac{2}{x+\sqrt{x^{2}+1}} \times\left(1+\frac{2 x}{2 \sqrt{x^{2}+1}}\right)\\ &\frac{\left(x^{2}+1\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}}{\sqrt{x^{2}+1}}=\frac{2}{x+\sqrt{x^{2}+1}} \times \frac{x+\sqrt{x^{2}+1}}{\sqrt{x^{2}+1}}\\ &\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=2 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 28 maths

Answer:
Proved
Hint:
You must know the derivative of logarithm and tangent inverse $x$
Given:
$y=\left(\tan ^{-1} x\right)^{2}, \text { show }\left(1+x^{2}\right)^{2} y_{2}+2 x\left(1+x^{2}\right) y_{1}=2$
Solution:
$\begin{aligned} &y=\left(\tan ^{-1} x\right)^{2}\\ &\text { Differentiate the equation w.r.t } x\\ &\frac{d y}{d x}=2 \tan ^{-1} x \cdot \frac{1}{1+x^{2}}\\ &\left(1+x^{2}\right) \frac{d y}{d x}=2 \tan ^{-1} x \end{aligned}$
$\begin{aligned} &\text { Differentiate again }\\ &\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+(2 x) \frac{d y}{d x}=\frac{2}{1+x^{2}}\\ &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(1+x^{2}\right)(2 x)=2\\ or\\ &\left(1+x^{2}\right)^{2} y_{2}+2 x\left(1+x^{2}\right) y_{1}=2 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 29

Answer:
Proved
Hint:
You must know the derivative of cot x function
Given:
$y=\cot x, \text { show } \frac{d^{2} y}{d x^{2}}+(2 y) \frac{d y}{d x}=0$
Solution:
$Let\; \; y=\cot x$
$\begin{aligned} &\text { Differentiate the equation w.r.t } x\\ &\frac{d y}{d x}=\frac{d(\cot x)}{d x}\\ &\frac{d y}{d x}=-\operatorname{cosec}^{2} x \end{aligned}$
$\begin{aligned} &\text { Differentiate again }\\ &\frac{d^{2} y}{d x^{2}}=-[2 cosec\: x(-cosec\: x \cot x)]\\ &\frac{d^{2} y}{d x^{2}}=-2 \operatorname{cosec}^{2} x \cot x \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=-2 \frac{d y}{d x} y \\ &\frac{d^{2} y}{d x^{2}}+(2 y) \frac{d y}{d x}=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 30

Answer:
$\frac{-2}{x^{2}}$
Hint:
You must know the derivative of logarithm function
Given:
$y=\log \left(\frac{x^{2}}{e^{2}}\right), \text { find } \frac{d^{2} y}{d x^{2}}$
Solution:
$Let\: \: y=\log \left(\frac{x^{2}}{e^{2}}\right)$
$Di\! f\! f\! erentiate\; the\; equation\; w.r.t\; x \\ \frac{d y}{d x}=\frac{1}{\frac{x^{2}}{e^{2}}} \cdot \frac{1}{e^{2}} \cdot 2 x=\frac{2}{x}\\ Di\! f\! f\! erentiate\; again\\\\ \frac{d^{2} y}{d x^{2}}=-2\left[\frac{1}{x^{2}}\right]=\frac{-2}{x^{2}}\\ Hence, \frac{d^{2} y}{d x^{2}}=\frac{-2}{x^{2}}$

Higher Order Derivatives exercise 11.1 question 31

Answer:
Proved
Hint:
You must know the derivative of exponential function
Given:
$y=a e^{2 x}+b e^{-x}, \text { show } \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-2 y=0$
Solution:
$y=a e^{2 x}+b e^{-x}$
$\begin{aligned} &\text { Differentiate the equation w.r.t } x\\ &\frac{d y}{d x}=2 a e^{2 x}+b e^{-x}\\ &\frac{d^{2} y}{d x^{2}}=4 a e^{2 x}+b e^{-x} \end{aligned}$
$\begin{aligned} &\text { LHS }=\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-2 y \\ &=4 a e^{2 x}+b e^{-x}-2 a e^{2 x}+b e^{-x} \\ &=-2 a e^{2 x}-2 b e^{2 x} \\ &=0=\text { RHS } \end{aligned}$

Higher Order Derivatives exercise 11.1 question 31

Higher Order Derivatives exercise 11.1 question 32 maths

Answer:
Proved
Hint:
You must know the derivative of exponential sin and cos functions
Given:
$y=e^{x}(\sin x+\cos x), \text { show } \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$
Solution:
$y=e^{x}(\sin x+\cos x)$
$\begin{aligned} &\text { Differentiate the equation w.r.t } x\\ &\frac{d y}{d x}=e^{x}(\cos x-\sin x)+e^{x}(\sin x+\cos x)\\ &\frac{d y}{d x}=e^{x}(\cos x-\sin x)+y \end{aligned}$
$\begin{aligned} &\text { Differentiate again }\\ &\frac{d^{2} y}{d x^{2}}=e^{x}(-\sin x-\cos x)+e^{x}(\cos x-\sin x)+\frac{d y}{d x}\\ &\frac{d^{2} y}{d x^{2}}=-y+\frac{d y}{d x}-y+\frac{d y}{d x}\\ &\therefore \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 33

Answer:
$-cot\: y.cosec^{2}y$
Hint:
You must know the derivative of cos inverse function
Given:
$y=\cos ^{-1} x, \text { find } \frac{d^{2} y}{d x^{2}} \text { in terms of } y \text { alone }$
Solution:
$y=\cos ^{-1} x$
$\begin{aligned} &\text { Differentiate the equation w.r.t } x\\ &\frac{d y}{d x}=\frac{d\left(\cos ^{-1} x\right)}{d x}\\ &=\frac{-1}{\sqrt{1-x^{2}}}=-\left(1-x^{2}\right)^{\frac{-1}{2}}\\ &\frac{d^{2} y}{d x^{2}}=\frac{d\left[-\left(1-x^{2}\right)^{\frac{-1}{2}}\right]}{d x} \end{aligned}$
$\begin{aligned} &=-\left(\frac{-1}{2}\right)\left(1-x^{2}\right)^{\frac{-3}{2}} \times(-2 x) \\ &=\frac{1}{2 \sqrt{\left(1-x^{2}\right)^{3}}} \times(-2 x) \\ &\frac{d^{2} y}{d x^{2}}=\frac{-x}{\sqrt{\left(1-x^{2}\right)^{3}}} \end{aligned}$
$\begin{aligned} &y=\cos ^{-1} x\\ &x=\cos y\\ &\text { Put in above equation }\\ &\frac{d^{2} y}{d x^{2}}=\frac{-\cos y}{\sqrt{\left(1-\cos ^{2} y\right)^{3}}} \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{-\cos y}{\sqrt{\sin ^{3} y}} \\ &=\frac{-\cos y}{\sin ^{3} y} \\ &=\frac{-\cos y}{\sin y} \times \frac{1}{\sin ^{2} y} \\ &\therefore \frac{d^{2} y}{d x^{2}}=-\cot y \cdot \operatorname{cosec}^{2} y \end{aligned}$

Higher Order Derivatives exercise 11.1 question 34

Answer:
Proved
Hint:
You must know the derivative of exponential and cos inverse function
Given:
$y=e^{a \cos ^{-1} x} , \text { prove }\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0$
Solution:
$y=e^{a \cos ^{-1} x}$
$\begin{aligned} &\text { Taking logarithm both sides }\\ &\log y=a \cos ^{-1} x \log c\\ &\log y=a \cos ^{-1} x\\ &\frac{1}{y} \frac{d y}{d x}=a \times\left(\frac{-1}{\sqrt{1-x^{2}}}\right)\\ &\frac{d y}{d x}=\frac{a y}{\sqrt{1-x^{2}}} \end{aligned}$
$\begin{aligned} &\text { Squaring both sides, }\\ &\left(\frac{d y}{d x}\right)^{2}=\frac{a^{2} y^{2}}{\left(1-x^{2}\right)}\\ &\left(1-x^{2}\right)\left(\frac{d y}{d x}\right)^{2}=a^{2} y^{2} \end{aligned}$
$\begin{aligned} &\text { Again differentiate, }\\ &\left(\frac{d y}{d x}\right)^{2}(-2 x)+\left(1-x^{2}\right) \times 2 \frac{d y}{d x} \frac{d^{2} y}{d x^{2}}=a^{2} \cdot 2 y \cdot \frac{d y}{d x}\\ &-x \frac{d y}{d x}+\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}=a^{2} y\\ &\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 35

Answer:
Proved
HInt:
You must know the derivative of exponential function
Given:
$y=500 e^{7 x}+600 e^{-7 x}, \text { show } \frac{d^{2} y}{d x^{2}}=49 y$
Solution:
$y=500 e^{7 x}+600 e^{-7 x}$
$\begin{aligned} &\frac{d y}{d x}=(7)(500) e^{7 x}+(-7)(600) e^{-7 x} \\ &\frac{d^{2} y}{d x^{2}}=(7)(7)(500) e^{7 x}+(-7)(-7)(600) e^{-7 x} \\ &\frac{d^{2} y}{d x^{2}}=(49)(500) e^{7 x}+(49)(600) e^{-7 x} \\ &\frac{d^{2} y}{d x^{2}}=(49)\left[(500) e^{7 x}+(600) e^{-7 x}\right] \\ &\frac{d^{2} y}{d x^{2}}=49 y \end{aligned}$

Higher Order Derivatives exercise 11.1 question 36 maths

Answer:
$\frac{-3}{2}$
Hint:
You must know the derivative of cos and sin function
$\begin{aligned} &x=2 \cos t-\cos 2 t \\ \end{aligned}$
Given:
$\begin{aligned} &y=2 \sin t-\sin 2 t , \text { find } \frac{d^{2} y}{d x^{2}} \text { at } t=\frac{\pi}{2} \end{aligned}$
Solution:
$\begin{aligned} &x=2 \cos t-\cos 2 t \\ &y=2 \sin t-\sin 2 t \\ &\frac{d x}{d t}=-2 \sin t+2 \sin 2 t \\ &\frac{d y}{d t}=2 \cos t-2 \cos 2 t \end{aligned}$
$\begin{aligned} &\text { Now, }\\ &\frac{d y}{d x}=\frac{2 \cos t-2 \cos 2 t}{-2 \sin t+2 \sin 2 t} \Rightarrow \frac{\cos t-\cos 2 t}{\sin 2 t-\sin t}\\ &=\frac{2 \sin \frac{3 t}{2} \sin \frac{t}{2}}{2 \cos \frac{3 t}{2} \sin \frac{t}{2}} \Rightarrow \tan \frac{3 t}{2} \end{aligned}$
$\begin{aligned} &\text { Therefore, }\\ &\frac{d^{2} y}{d x^{2}}=\sec ^{2} \frac{3 t}{2} \times \frac{3}{2} \times \frac{d t}{d x}\\ &=3 \sec ^{2} \frac{3 t}{2} \cdot \frac{1}{2 \sin 2 t-2 \sin t}\\ &\left.\frac{d^{2} y}{d x^{2}}\right]_{t=\frac{\pi}{2}}=\frac{-3}{2} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 37

Answer:
$\frac{-7}{64z^{3}}$
Hint:
You must know the derivative of $x$ and $y$
Given:
$\begin{aligned} &x=4 z^{2}+5 \\ &y=6 z^{2}+7 z+3 , \text { find } \frac{d^{2} y}{d x^{2}} \end{aligned}$
Solution:
$\begin{aligned} &x=4 z^{2}+5\\ &y=6 z^{2}+7 z+3\\ &\frac{d x}{d z}=8 z+0=8 z\\ &\frac{d y}{d z}=12 z+7\\ &\text { Now, }\\ &\frac{d y}{d x}=\frac{12 z+7}{8 z}\\ &\frac{d y}{d x}=\frac{3}{2}+\frac{7}{8 z} \end{aligned}$
$Again\; di\! f\!\! f\! erentiating\; w.r.t\; z \\ \frac{d^{2} y}{d x^{2}} \times \frac{d x}{d z}=-\frac{7}{8 z^{2}}\\ \\ or\\ \\ \frac{d^{2} y}{d x^{2}}=-\frac{7}{8 z^{2}} \times \frac{d z}{d x}\\ \\ =-\frac{7}{8 z^{2}} \times \frac{1}{8 z}\\ \frac{d^{2} y}{d x^{2}}=\frac{-7}{64 z^{3}}$

Higher Order Derivatives exercise 11.1 question 38

Answer:
Proved
Hint:
You must know the derivative of logarithm and cos function
Given:
$y=\log (1+\cos x) , \text { prove } \frac{d^{3} y}{d x^{3}}+\frac{d^{2} y}{d x^{2}} \times \frac{d y}{d x}=0$
Solution:
$y=\log (1+\cos x)$
$\begin{aligned} &\frac{d y}{d x}=\frac{-\sin x}{1+\cos x}\\ &\frac{d^{2} y}{d x^{2}}=\frac{-\cos x-\cos ^{2} x-\sin ^{2} x}{(1+\cos x)^{2}}\\ &=\frac{-(\cos x+1)}{(1+\cos x)}\\ &=\frac{-1}{1+\cos x}\\ &\text { Again differentiating } \end{aligned}$
$\begin{aligned} &\frac{d^{3} y}{d x^{3}}=\frac{-\sin x}{(1+\cos x)^{2}} \\ &\frac{d^{3} y}{d x^{3}}+\frac{\sin x}{(1+\cos x)^{2}}=0 \\ &\frac{d^{3} y}{d x^{3}}+\left(\frac{-1}{1+\cos x}\right)\left(\frac{-\sin x}{1+\cos x}\right)=0 \\ &\frac{d^{3} y}{d x^{3}}+\frac{d^{2} y}{d x^{2}} \times \frac{d y}{d x}=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 39

Answer:
Proved
Hint:
You must know the derivative of sin and logarithm function
Given:
$y=\sin (\log x) , \text { prove } x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0$
Solution:
$y=\sin (\log x)$
$\begin{aligned} &\frac{d y}{d x}=\cos (\log x) \times \frac{1}{x}\\ &=\frac{\cos (\log x)}{x}\\ &\text { Again differentiating }\\ &\frac{d^{2} y}{d x^{2}}=\frac{x\left[-\sin (\log x) \times \frac{1}{x}\right]-\cos (\log x)}{x^{2}}\\ &=\frac{-\cos (\log x)-\sin (\log x)}{x^{2}} \end{aligned}$
$\begin{aligned} &\text { Now, }\\ &\text { LHS }=x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y\\ &=\frac{x^{2}\{-\cos (\log x)-\sin (\log x)\}}{x^{2}}+\frac{x \cos (\log x)}{x}+\sin (\log x)\\ &=0=\mathrm{RHS} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 40 maths

Answer:
Proved
Hint:
You must know the derivative of exponential function
Given:
$y=3 e^{2 x}+2 e^{3 x}, \text { prove } \frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0$
Solution:
$y=3 e^{2 x}+2 e^{3 x}$
$\begin{aligned} &\frac{d y}{d x}=(2)(3) e^{2 x}+(3)(2) e^{3 x} \\ &=6 e^{2 x}+6 e^{3 x} \\ &\frac{d y}{d x}=6 e^{2 x}+\frac{6\left(y-3 e^{2 x}\right)}{2} \\ &\frac{d y}{d x}=6 e^{2 x}+3 y-9 e^{2 x} \\ &=-3 e^{2 x}+3 y \end{aligned}$
$\begin{aligned} &\text { Again differentiating }\\ &\frac{d^{2} y}{d x^{2}}=\frac{3 d y}{d x}-6 e^{2 x} \quad \ldots \ldots \ldots .(1)\\ &\frac{d y}{d x}-3 y=-3 e^{2 x}\\ &\frac{\frac{d y}{d x}-3 y}{-3}=e^{2 x} \end{aligned}$
$\begin{aligned} &\text { Put in (1), }\\ &\frac{d^{2} y}{d x^{2}}=\frac{3 d y}{d x}-6\left(\frac{\frac{d y}{d x}-3 y}{-3}\right)\\ &\frac{d^{2} y}{d x^{2}}=\frac{3 d y}{d x}+2 \frac{d y}{d x}-6 y\\ &\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 41

Answer:
Proved
Hint:
You must know the derivative of cot inverse $x$
Given:
$y=\left(\cot ^{-1} x\right)^{2}, \text { prove } y_{2}\left(x^{2}+1\right)^{2}+2 x\left(x^{2}+1\right) y_{1}=2$
Solution:
$y=\left(\cot ^{-1} x\right)^{2}$
$\begin{aligned} &\frac{d y}{d x}=2 \cot ^{-1} x\left(\frac{-1}{1+x^{2}}\right)\\ &\left(1+x^{2}\right) \frac{d y}{d x}=-2 \cot ^{-1} x\\ &\text { Again differentiating }\\ &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=-2\left(\frac{-1}{1+x^{2}}\right) \end{aligned}$
$\begin{aligned} &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=\left(\frac{2}{1+x^{2}}\right) \\ &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2 \\ &y_{2}\left(x^{2}+1\right)^{2}+2 x\left(x^{2}+1\right) y_{1}=2 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 42

Answer:
Proved
Hint:
You must know the derivative of cosec^-1x
Given:
$y=\left(\operatorname{cosec}^{-1} x\right), x>1 , \text { prove } x\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+\left(2 x^{2}-1\right) \frac{d y}{d x}=0$
Solution:
$\begin{aligned} &y=\left(\cos e c^{-1} x\right) \\ &\frac{d y}{d x}=\frac{-1}{x \sqrt{x^{2}-1}} \\ &x \sqrt{x^{2}-1} \frac{d y}{d x}=-1 \end{aligned}$
$\begin{aligned} &\text { Again differentiating }\\ &x \sqrt{x^{2}-1} \frac{d^{2} y}{d x^{2}}+\sqrt{x^{2}-1} \frac{d y}{d x}+x \cdot \frac{2 x}{2 \sqrt{x^{2}-1}} \frac{d y}{d x}=0\\ &x\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+\left(2 x^{2}-1\right) \frac{d y}{d x}=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 43

Answer:
$2 \sqrt{2}=\frac{d^{2} y}{d x^{2}} \text { and } \frac{-1}{\sqrt{x}}=\frac{d^{2} y}{d x^{2}}$
Hint:
You must know the derivative of cos, tan, sin and logarithm function
Given:
$\begin{aligned} &x=\cos t+\log \tan \frac{t}{2} \\ &y=\sin t \quad \text { find } \frac{d^{2} y}{d x^{2}} \& \frac{d^{2} y}{d x^{2}} \quad \text { at } t=\frac{\pi}{4} \end{aligned}$
Solution:
$\begin{aligned} &y=\sin t \quad \frac{d y}{d t}=\cos t \\ &\frac{d^{2} y}{d t^{2}}=-\sin t \\ &\left.\frac{d^{2} y}{d t^{2}}\right]_{t=\frac{\pi}{4}}=-\sin \frac{\pi}{4}=\frac{-1}{\sqrt{2}} \end{aligned}$
$\begin{aligned} &\text { Again differentiating }\\ &x=\cos t+\log \tan \frac{t}{2}\\ &\frac{d x}{d t}=-\sin t+\frac{1}{\tan \frac{t}{2}} \cdot \sec ^{2} \frac{t}{2} \cdot \frac{1}{2}\\ &=-\sin t+\frac{\cos \frac{t}{2}}{2 \times \sin \frac{t}{2}} \cdot \sec ^{2} \frac{t}{2} \cdot \frac{1}{2}\\ &=-\sin t+\frac{1}{\sin 2 \times \frac{t}{2}} \end{aligned}$
$\begin{aligned} &=-\sin t+\operatorname{cosec\: t} \\ &\text { Now }, \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{\cos t}{\operatorname{cosec\: t}-\sin t} \\ &=\frac{\cos t}{1-\sin ^{2} t} \sin t=\frac{\sin t \cos t}{\cos ^{2} t} \\ &\frac{d^{2} y}{d x^{2}}=\frac{d\left(\frac{d y}{d x}\right)}{d x} \end{aligned}$
$\begin{aligned} &=\frac{\frac{d}{d t}\left(\frac{d y}{d x}\right)}{\frac{d x}{d t}}=\frac{\sec ^{2} t}{\operatorname{cosec\: t}-\sin t} \\ &=\frac{\sec ^{2} t \cdot \sin t}{\cos ^{2} t}=\sec ^{2} t \tan t \\ &\left.\frac{d^{2} y}{d x^{2}}\right]_{-\frac{\pi}{4}}=2 \sqrt{2} \times 1 \\ &=2 \sqrt{2} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 44 maths

Answer:
$\frac{1}{a \sin ^{2} t \cos t}$
Hint:
You must know the derivative of sin, cos, tan and logarithm function
Given:
$\begin{aligned} &x=a \sin t \\ &y=a\left(\cos t+\log \tan \frac{t}{2}\right) \text { find } \frac{d^{2} y}{d x^{2}} \end{aligned}$
Solution:
$\begin{aligned} &x=a \sin t \\ &\frac{d x}{d t}=a \cos t \\ &y=a\left(\cos t+\log \tan \frac{t}{2}\right) \\ &\frac{d y}{d t}=a\left(-\sin t+\cot \frac{t}{2} \times \sec ^{2} \frac{t}{2} \times \frac{t}{2}\right) \end{aligned}$
$\begin{aligned} &=a\left(-\sin t+\frac{1}{2 \sin \left(\frac{t}{2}\right) \times \cos \left(\frac{t}{2}\right)}\right) \\ &\frac{d y}{d t}=a\left(-\sin t+\frac{1}{\sin t}\right) \\ &=a\left(\frac{-\sin ^{2} t+1}{\sin t}\right) \Rightarrow \frac{a \cos ^{2} t}{\sin t} \end{aligned}$
$\begin{aligned} &\therefore \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{\frac{a \cos ^{2} t}{\sin t}}{a \cos t} \\ &=\frac{\cos t}{\sin t}=\cot t \\ &\text { Again } \frac{d^{2} y}{d x^{2}}=-cosec^{2} t \frac{d t}{d x}=cose c^{2} t \times \frac{1}{a \cos t} \\ &=\frac{1}{a \sin ^{2} t \cos t} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 45

Answer:
$\frac{8\sqrt{2}}{\pi a}$
Hint:
You must know the derivative of cos and sin function
Given:
$\begin{aligned} &x=a(\cos t+t \sin t) \\ &y=a(\sin t-t \cos t) \text { find } \frac{d^{2} y}{d x^{2}} \text { at } t=\frac{\pi}{4} \end{aligned}$
Solution:
$\begin{aligned} &x=a(\cos t+t \sin t) \\ &\frac{d x}{d t}=a(-\sin t+t \cos t+\sin t) \\ &=a t \cos t \\ &y=a(\sin t-t \cos t) \\ &\frac{d y}{d t}=a(\cos t+t \sin t-\cos t) \\ &=a t \sin t \end{aligned}$
$\begin{aligned} &\therefore \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{a t \sin t}{a t \cos t}=\tan t \\ &\text { Again } \frac{d^{2} y}{d x^{2}}=\sec ^{2} t \frac{d t}{d x}=\sec ^{2} t \times \frac{1}{a t \cos t} \end{aligned}$
$\begin{aligned} &=\frac{\sec ^{2} t}{a t} \\ &\left.\frac{d^{2} y}{d x^{2}}\right]_{t=\frac{\pi}{4}}=\frac{\sec ^{2} \frac{\pi}{4}}{a \frac{\pi}{4}}=\frac{2 \sqrt{2} \times 4}{a \pi} \\ &=\frac{8 \sqrt{2}}{\pi a} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 46

Answer:
$\frac{8\sqrt{3}}{a}$
Hint:
You must know the derivative of cos, sin, tan and logarithm function
Given:
$\begin{aligned} &x=a\left(\cos t+\log \tan \frac{t}{2}\right) \\ &y=a \sin t \quad \text { find } \frac{d^{2} y}{d x^{2}} \text { at } t=\frac{\pi}{3} \end{aligned}$
Solution:
$\begin{aligned} &x=a\left(\cos t+\log \tan \frac{t}{2}\right) \\ &\frac{d x}{d t}=a\left[-\sin t+\frac{1}{\tan \frac{t}{2}} \sec ^{2} \frac{t}{2} \cdot \frac{1}{2}\right] \\ &\frac{d x}{d t}=a\left[-\sin t+\frac{\cos \frac{t}{2}}{2 \sin \frac{t}{2}} \frac{1}{\cos ^{2} \frac{t}{2}}\right] \end{aligned}$
$\begin{aligned} &\frac{d x}{d t}=a\left[-\sin t+\frac{1}{2 \sin \frac{t}{2} \cos \frac{t}{2}}\right] \\ &\frac{d x}{d t}=a\left[-\sin t+\frac{1}{\sin t}\right] \\ &\frac{d x}{d t}=\frac{a \cos ^{2} t}{\sin t} \\ &y=a \sin t \end{aligned}$
$\begin{aligned} &\frac{d y}{d t}=a \cos t \\ &\therefore \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{a \cos t}{a \cos ^{2} t} \times \sin t=\tan t \\ &\text { Again } \frac{d^{2} y}{d x^{2}}=\sec ^{2} t \times \frac{\sin t}{a \cos ^{2} t} \end{aligned}$
$\begin{aligned} &\left.\frac{d^{2} y}{d x^{2}}\right]_{t=\frac{\pi}{3}}=\frac{\sec ^{2} \frac{\pi}{3} \cdot \sin \frac{\pi}{3}}{a \cos ^{2} \frac{\pi}{4}}=\frac{(2)^{2} \cdot\left(\frac{\sqrt{3}}{2}\right)}{a\left(\frac{1}{2}\right)^{2}} \\ &\left.\frac{d^{2} y}{d x^{2}}\right]_{t=\frac{\pi}{3}}=\frac{8 \sqrt{3}}{a} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 47

Answer:
$\frac{(\sin 2 t+2 a t \cos 2 t)}{(\cos 2 t-2 a t \sin 2 t)}$
Hint:
You must know the derivative of cos and sin function
Given:
$\begin{aligned} &x=a(\cos 2 t+2 t \sin 2 t) \\ &y=a(\sin 2 t-2 t \cos 2 t) \quad \text { find } \frac{d^{2} y}{d x^{2}} \end{aligned}$
Solution:
$\begin{aligned} &x=a(\cos 2 t+2 t \sin 2 t) \\ &\frac{d x}{d t}=a(-2 \sin 2 t+2 \sin 2 t+4 t \cos 2 t) \\ &=4 a t \cos 2 t \\ &\frac{d^{2} x}{d t^{2}}=4 a \cos 2 t-8 a t \sin 2 t \end{aligned}$
$\begin{aligned} &y=a(\sin 2 t-2 t \cos 2 t) \\ &\frac{d y}{d t}=a(2 \cos 2 t-2 \cos 2 t+4 t \sin 2 t) \\ &=4 a t \sin 2 t \\ &\frac{d^{2} y}{d t^{2}}=4 a \sin 2 t+8 a t \cos 2 t \end{aligned}$
$\begin{aligned} &\text { So, } \frac{d^{2} y}{d x^{2}}=\frac{4 \sin 2 t+8 a t \cos 2 t}{4 \cos 2 t-8 a t \sin 2 t} \\ &=\frac{4(\sin 2 t+2 a t \cos 2 t)}{4(\cos 2 t-2 a t \sin 2 t)} \\ &=\frac{(\sin 2 t+2 a t \cos 2 t)}{(\cos 2 t-2 a t \sin 2 t)} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 48 maths

Answer:
$\frac{-1}{3 \sin ^{3} t \cos 2 t}$
Hint:
You must know the derivative of cos t and sin t function
Given:
$\begin{aligned} &x=3 \cos t-2 \cos ^{3} t \\ &y=3 \sin t-2 \sin ^{3} t \quad \text { find } \frac{d^{2} y}{d x^{2}} \end{aligned}$
Solution:
$\begin{aligned} &x=3 \cos t-2 \cos ^{3} t \\ &\frac{d x}{d t}=-3 \sin t+6 \cos ^{2} t \sin t \\ &=\sin t+\left(6 \cos ^{2} t-3\right) \\ &y=3 \sin t-2 \sin ^{3} t \\ &\frac{d y}{d t}=3 \cos t-6 \sin ^{2} t \cos t \\ &=\cos t\left(3-6 \sin ^{2} t\right) \end{aligned}$
$\begin{aligned} &\therefore \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d x}{d t}=\frac{\cos t\left(3-6 \sin ^{2} t\right)}{\sin t+\left(6 \cos ^{2} t-3\right)} \\ &=\frac{\cos t\left(1-2 \sin ^{2} t\right)}{3\left(2 \cos ^{2} t-1\right)} \\ &=\frac{\cot t(\cos 2 t)}{\cos 2 t}=\cot t \\ &\frac{d y}{d x}=\cot t \\ &\frac{d^{2} y}{d x^{2}}=-cos e c^{2} t \end{aligned}$

Higher Order Derivatives exercise 11.1 question 49

Answer:
$\frac{-(x^{2}+y^{2})}{y^{3}}$
Hint:
You must know the derivative of cos t and sin t function
Given:
$\begin{aligned} &x=a \sin t-b \cos t \\ &y=a \cos t-b \sin t \quad \text { Prove } \frac{d^{2} y}{d x^{2}}=\frac{-\left(x^{2}+y^{2}\right)}{y^{3}} \end{aligned}$
Solution:
$\begin{aligned} &\frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{a \sin t+b \cos t}{a \cos t+b \sin t}\\ &\text { Use quotient rule }\\ &\frac{U}{V}=\frac{U V^{\prime}-U^{\prime} V}{V^{2}} \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{\frac{d\left(\frac{d y}{d x}\right)}{d t}}{\frac{d x}{d t}}\\ &=\frac{(-a \cos t-b \sin t)(a \cos t+b \sin t)-(a \sin t+b \cos t)(-a \sin t+b \cos t)}{(a \cos t+b \sin t)^{2}(a \cos t+b \sin t)}\\ &=\frac{(-a \cos t+b \sin t)^{2}-(b \cos t-a \sin t)^{2}}{y^{3}}\\ &=\frac{-y^{2}-x^{2}}{y^{3}}=\frac{-\left(x^{2}+y^{2}\right)}{y^{3}} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 50

Answer:
$A=\frac{2}{3}\: and\: B=\frac{-1}{3}$
Hint:
You must know the derivative of second order
Given:
$\text { Find } A \& B \text { so that } y=A \sin 3 x+B \cos 3 x \text { satisfies the equation } \frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+3 y=10 \cos 3 x$
Solution:
$\begin{aligned} &\text { Let } y=A \sin 3 x+B \cos 3 x \\ &\frac{d y}{d x}=\frac{d(A \sin 3 x+B \cos 3 x)}{d x} \\ &=A \cos 3 x-3+b(-\sin 3 x .3) \quad\left[\begin{array}{l} \frac{d \cos 3 x}{d x}=-3 \sin 3 x \\ \frac{d \sin 3 x}{d x}=3 \cos 3 x \end{array}\right] \end{aligned}$
$\begin{aligned} &\frac{d y}{d x}=3 A \cos 3 x-3 B \sin 3 x \\ &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}(3 A \cos 3 x-3 B \sin 3 x) \\ &=3 A(-\sin 3 x \cdot 3)-3 B(\cos 3 x .3) \\ &=-9 A \sin 3 x-9 B \cos 3 x \\ &=-9(A \sin 3 x+3 \cos 3 x) \\ &=-9 y \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+3 y=10 \cos 3 x\\ &-9 y+4(3 A \cos 3 x-3 B \sin 3 x)+3 y=10 \cos 3 x\\ &-6 y+12 A \cos 3 x-12 B \sin 3 x=10 \cos 3 x\\ &-6(A \sin 3 x+B \cos 3 x)+12 A \cos 3 x-12 B \sin 3 x=10 \cos 3 x\\ &\sin 3 x(-6 A-12 B)+\cos 3 x(-6 B+12 A)=10 \cos 3 x\\ &-6 A-12 B=0 \quad \ldots \ldots . .(1)\\ &-6 B+12 A=0 \quad \ldots \ldots . .(2)\\ &6 A=-12 B\\ &A=-2 B \quad \ldots \ldots(3) \end{aligned}$
$\begin{aligned} &\text { Put (3) in (2) }\\ &-6 B+(-2 B) 12=10\\ &-6 B-24 B=10\\ &-30 B=10\\ &B=\frac{-1}{3}\\ &A=-2\left(\frac{-1}{3}\right)=\frac{2}{3}\\ &A=\frac{2}{3}\; \&\; B=\frac{-1}{3} \end{aligned}$

Higher Order Derivatives exercise 11.1 question 51

Answer:
$\frac{d^{2} y}{d t^{2}}+2 k \frac{d y}{d t}+n^{2} y=0$
Hint:
You must know the derivative of
$A e^{-k t} \cos (p t+c)$
Given:
$\text { If } y=A e^{k t} \cos (p t+c) \text { prove that } \frac{d^{2} y}{d t^{2}}+2 k \frac{d y}{d t}+n^{2} y=0 \text { where } n^{2}=p^{2}+k^{2}$
Solution:
$\begin{aligned} &\text { Let } y=A e^{-k t} \cos (p t+c) \quad \ldots . .(1) \\ &\frac{d y}{d x}=A e^{-k t}(-k) \cos (p t+c)+A e^{-k t}(-\sin (p t+c) p) \\ &\frac{d y}{d t}=-A k e^{-k t} \cos (p t+c)-A p e^{-k t}(\sin (p t+c)) \quad \ldots . .(2) \end{aligned}$
$\begin{aligned} &\frac{d^{2} y}{d x^{2}}=A k^{2} e^{-k t} \cos (p t+c)+A p k e^{-k t} \sin (p t+c)-A p k e^{-k t} \sin (p t+c)-A p^{2} e^{-k t} \cos (p t+c) \\ &\frac{d^{2} y}{d t^{2}}=\left(k^{2}-p^{2}\right) A e^{-k t} \cos (p t+c)+2 k\left(A p e^{-k t} \sin (p t+c)\right) \quad \ldots . .(3) \end{aligned}$
$\begin{aligned} &\text { Using }(1)\: \&\: (2)\: \&\: n^{2}=k^{2}+p^{2} \\ &\frac{d^{2} y}{d t^{2}}=\left(k^{2}-p^{2}\right) y+2 k\left(-k y-\frac{d y}{d t}\right) \\ &\frac{d^{2} y}{d t^{2}}=\left(k^{2}-p^{2}-2 k^{2}\right) y+2 k\left(\frac{d y}{d t}\right) \\ &\frac{d^{2} y}{d t^{2}}=-n^{2} y-2 k \frac{d y}{d t} \\ &\frac{d^{2} y}{d t^{2}}+2 k \frac{d y}{d t}+n^{2} y=0 \end{aligned}$

Higher Order Derivatives exercise 11.1 question 52 maths

Answer:
$x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0$
Hint:
You must know the derivative of
$cos(log\: x)\: and\: sin(log\: x)$
Given:
$\text { If } y=x^{n}\{\operatorname{acos}(\log x)+b \sin (\log x)\} \text { prove that } x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0$
Solution:
$\begin{aligned} &\text { Let } y=x^{n}\{a \cos (\log x)+b \sin (\log x)\}\\ &\text { Use multiplicative rule }\\ &\mathrm{UV}=\mathrm{U}^{\prime} \mathrm{V}+\mathrm{UV}^{\prime}\\ &U=x^{n}\\ &V=a \cos (\log x)+b \sin (\log x) \end{aligned}$
$\begin{aligned} &\frac{d y}{d x}=n x^{n-1}\{\operatorname{acos}(\log x)+b \sin (\log x)\}+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \\ &\frac{d y}{d x}=n \frac{y}{x}+\frac{x^{n}}{x}(-a \sin (\log x)+b \cos (\log x)) \\ &x \frac{d y}{d x}=n y+x^{n}(-a \sin (\log x)+b \cos (\log x)) \end{aligned}$
$\begin{aligned} &\frac{d y}{d x}+x \frac{d^{2} y}{d x^{2}}=n \frac{d y}{d x}+n x^{n-1}(-a \sin (\log x)+b \cos (\log x))+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \\ &\frac{d y}{d x}(1-x)+x \frac{d^{2} y}{d x^{2}}=n x^{n-1}(-a \sin (\log x)+b \cos (\log x))+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \end{aligned}$
$\begin{aligned} &\frac{d y}{d x}(1-n)+x \frac{d^{2} y}{d x^{2}}=n x^{n-1}(-a \sin (\log x)+b \cos (\log x))-\frac{1}{x} y \\ &\frac{d y}{d x}(1-n)+x \frac{d^{2} y}{d x^{2}}=\frac{n}{x}\left(x \frac{d y}{d x}-n y\right)-\frac{y}{x} \end{aligned}$
$\begin{aligned} &\frac{d y}{d x}+x(1-n) \frac{d y}{d x}=n x \frac{d y}{d x}-n^{2} y-y \\ &\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}(x-n x-n x)-\left(1+n^{2}\right) y=0 \\ &x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0 \end{aligned}$

The RD Sharma Class 12 Solutions Higher Order Derivatives Ex 11.1 is available for the students who find difficulty in this chapter. The 11th chapter of class 12 mathematics consists of only 1 exercise, ex 11.1. Therefore, it is the smallest and one of the easiest chapters that is present in the syllabus.

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