RD Sharma materials are well known among Indian schools as they are one of the best materials for maths. These materials also surpass NCERT in terms of chapter details and exam-oriented questions.
RD Sharma Class 12th Exercise 10.6 contains the concepts for Differentiation. This is a relatively small exercise containing eight questions only. These include Level 1 questions that have proof sums and evaluation. As the sums deal with the same theorems, students can solve a few problems for their understanding and then refer to RD Sharma solutions material as it is more convenient.
RD Sharma Class 12 Solutions Chapter 10 Differentiation - Other Exercise
Differentiation Excercise: 10.6
Differentiation exercise 10.6 question 1
Answer:
$\frac{d y}{d x}=\frac{1}{2 y-1}$Hint: The value of y is given as infinite series. If a term is deleted from an infinite series, it remains the same in this case.
Given:
$y=\sqrt{x+\sqrt{x+\sqrt{x+\ldots \ldots \ldots+\infty}}}$Solution:
Here it is given that,
$y=\sqrt{x+\sqrt{x+\sqrt{x+\ldots \ldots \ldots+\infty}}}$This can be written as:
$y=\sqrt{x+y}$Squaring on both sides, we get:
$y^{2}=x+y$ …(1)
Differentiating (1) w.r.t x,
$\begin{aligned} &2 y \frac{d y}{d x}=1+\frac{d y}{d x} \\ &\frac{d y}{d x}(2 y-1)=1 \\ &\therefore \frac{d y}{d x}=\frac{1}{2 y-1} \end{aligned}$Hence, it is proved.
Differentiation exercise 10.6 question 2
Answer:
$\frac{d y}{d x}=\frac{\sin x}{1-2 y}$Hint: The value of y is given as infinite series. If a term is deleted from an infinite series, it remains the same in this case.
Given:
$y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \ldots \ldots . .+\infty}}}$Solution:
Here it is given that,
$y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \ldots \ldots . .+\infty}}}$This can be written as:
$y=\sqrt{\cos x+y}$Squaring on both sides, we get:
$y^{2}=\left ( \cos x+y \right )$ …(1)
Differentiating (1) w.r.t x,
$\begin{aligned} &2 y \frac{d y}{d x}=-\sin x+\frac{d y}{d x} \\ &\frac{d y}{d x}(2 y-1)=-\sin x \\ &\frac{d y}{d x}=\frac{-\sin x}{2 y-1} \end{aligned}$$\therefore \frac{d y}{d x}=\frac{-\sin x}{2 y-1}$Hence, it is proved.
Differentiation exercise 10.6 question 3
Answer:
$\left ( 2y-1 \right )\frac{dy}{dx}=\frac{1}{x}$Hint: The value of y is given as infinite series. If a term is deleted from an infinite series, it remains the same in this case.
Given:
$y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\ldots \ldots \ldots .+\infty}}}$Solution:
Here it is given that,
$y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\ldots \ldots \ldots .+\infty}}}$This can be written as:
$y=\sqrt{\log x+y}$Squaring on both sides, we get:
$y^{2}=\left ( \log x+y \right )$ …(1)
Differentiating (1) w.r.t x,
$\begin{aligned} &2 y \frac{d y}{d x}=\frac{1}{x}+\frac{d y}{d x} \\ &\frac{d y}{d x}(2 y-1)=\frac{1}{x} \end{aligned}$$\therefore \frac{d y}{d x}(2 y-1)=\frac{1}{x}$. Hence, it is proved.
Differentiation exercise 10.6 question 4
Answer:
$\frac{dy}{dx}=\frac{\sec ^{2}x}{2y-1}$Hint: The value of y is given as infinite series. If a term is deleted from an infinite series, it remains the same in this case.
Given:
$y=\sqrt{\tan x+\sqrt{\tan x+\sqrt{\tan x+\ldots \ldots \ldots .+\infty}}}$Solution:
Here it is given that,
$y=\sqrt{\tan x+\sqrt{\tan x+\sqrt{\tan x+\ldots \ldots \ldots .+\infty}}}$This can be written as:
$y=\sqrt{\tan x+y}$Squaring on both sides, we get:
$y^{2}=\left ( \tan x+y \right )$ …(1)
Differentiating (1) w.r.t x,
$2 y \frac{d y}{d x}=\sec ^{2} x+\frac{d y}{d x}$$\frac{d y}{d x}\left ( 2y-1 \right )=\sec ^{2} x$$\therefore \frac{d y}{d x}=\frac{\sec ^{2} x}{ 2y-1}$Hence, it is proved.
Differentiation exercise 10.6 question 5
Answer:
$\frac{d y}{d x}=\frac{y^{2} \cot x}{1-y \log \sin x}$Hint: The value of y is given as infinite series. If a term is deleted from an infinite series, it remains the same in this case.
Given:
$y=\left ( \sin x \right )^{\left ( \sin \right )^{..... \infty }}$Solution:
Here it is given that,
$y=\left ( \sin x \right )^{\left ( \sin \right )^{..... \infty }}$This can be written as:
$\log y=\log \left ( \sin x \right )^{y}$Taking log on both sides, we get:
$\log y=\log \left ( \sin x \right )^{y}$$\therefore \log y=y\log \left ( \sin x \right )$ …(1)
Differentiating (1) w.r.t x,
$\begin{aligned} &\frac{1}{y} \frac{d y}{d x}=y \frac{d}{d x}(\log \sin x)+\log \sin x \cdot \frac{d}{d x}(y) \\ &\frac{1}{y} \frac{d y}{d x}=y \frac{1}{\sin x}(\cos x)+\log \sin x \frac{d y}{d x} \\ &\frac{d y}{d x}\left(\frac{1}{y}-\log \sin x\right)=y \cot x \\ &\frac{d y}{d x}\left(\frac{1-y \log \sin x}{y}\right)=y \cot x \\ &\therefore \frac{d y}{d x}=\frac{y^{2} \cot x}{1-y \log \sin x} \end{aligned}$Hence, it is proved.
Differentiation exercise 10.6 question 6
Answer:
$\frac{dy}{dx}=2$Hint: The value of y is given as infinite series. If a term is deleted from an infinite series, it remains the same in this case.
Given:
$y=(\tan x)^{(\tan x)^{(\tan x) \cdot \cdots}}$Solution:
Here it is given that,
$y=(\tan x)^{(\tan x)^{(\tan x) \cdot \cdots}}$This can be written as:
$y=(\tan x)^{y}$Taking log on both sides, we get:
$\log y=\log (\tan x)^{y}$$\therefore \log y=y\log (\tan x)$ …(1)
Differentiating (1) w.r.t x,
$\begin{aligned} &\frac{1}{y} \frac{d y}{d x}=y \frac{d}{d x}(\log \tan x)+\log \tan x \cdot \frac{d}{d x}(y) \\ &\frac{1}{y} \frac{d y}{d x}=y \frac{1}{\tan x}\left(\sec ^{2} x\right)+\log \tan x \frac{d y}{d x} \\ &\frac{d y}{d x}\left(\frac{1}{y}-\log \tan x\right)=y \frac{\sec ^{2} x}{\tan x} \\ &\frac{d y}{d x}\left(\frac{1-y \log \tan x}{y}\right)=y \frac{\sec ^{2} x}{\tan x} \\ &\therefore \frac{d y}{d x}=\frac{y^{2}}{1-y \log \tan x} \cdot \frac{\sec ^{2} x}{\tan x} \end{aligned}$$\text { If } \mathrm{x}=\frac{\pi}{4}, \text { becomes: }$$\begin{aligned} \left(\frac{d y}{d x}\right)_{x=\left(\frac{\pi}{4}\right)} &=\frac{y^{2}}{1-y \log \tan \left(\frac{\pi}{4}\right)} \cdot \frac{\sec ^{2}\left(\frac{\pi}{4}\right)}{\tan \left(\frac{\pi}{4}\right)} \\ &=\frac{y^{2}(\sqrt{2})^{2}}{1-y \log 1} \\ &=\frac{2 y^{2}}{1-y(0)} \end{aligned}$$\begin{aligned} \text { Since, }(y) \frac{\pi}{4} &=\tan \left(\frac{\pi}{4}\right)^{\tan \left(\frac{\pi}{4}\right)^{\cdots-\infty}} \\ &=(1)^{\infty} \\ &=1 \end{aligned}$$\therefore y=1$$\frac{dy}{dx}=\frac{2\left ( 1 \right )^{2}}{1-0}$$\frac{dy}{dx}=2$Hence, it is proved.
Differentiation exercise 10.6 question 8
Answer:
$\frac{d y}{d x}=\frac{-y^{2} \tan x}{1-y \log \cos x}$Hint: The value of y is given as infinite series. If a term is deleted from an infinite series, it remains the same in this case.
Given:
$y=\left ( \cos x\right )^{\left ( \cos x \right )^{...\infty }}$Solution:
Here it is given that,
$y=\left ( \cos x\right )^{\left ( \cos x \right )^{...\infty }}$This can be written as:
$y=\left ( \cos x\right )^{y}$Taking log on both sides, we get:
$\begin{aligned} &\log y=\log (\cos x)^{y} \\ &\therefore \log y=y \log (\cos x) \end{aligned}$ …(1)
Differentiating (1) w.r.t x,
$\begin{aligned} &\frac{1}{y} \frac{d y}{d x}=y \frac{d}{d x}(\log \cos x)+\log \cos x \cdot \frac{d}{d x}(y) \\ &\frac{1}{y} \frac{d y}{d x}=y \frac{1}{\cos x}(-\sin x)+\log \cos x \frac{d y}{d x} \\ &\frac{d y}{d x}\left(\frac{1}{y}-\log \cos x\right)=-y \tan x \\ &\frac{d y}{d x}\left(\frac{1-y \log \cos x}{y}\right)=-y \tan x \\ &\therefore \frac{d y}{d x}=\frac{-y^{2} \tan x}{1-y \log \cos x} \end{aligned}$Most students mistake solving every single sum from the chapter even if they know the concepts. This is not the suggested method because instead of learning new concepts, they get stuck on the same sort of problem. Instead, students should practice solving until they understand the concept. Once that is done, they can refer to RD Sharma Class 12 Solutions Chapter 10 Ex 10.6 to verify similar problems. This method saves a lot of time and effort and helps them cover a significant portion of the syllabus.
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