CBSE Class 12th Exam Date:17 Feb' 26 - 17 Feb' 26
Continuity is like a peaceful river flowing without any gaps, and Differentiability is when each ripple of the river follows a certain pattern smoothly and precisely. Oftentimes, we come across some functions which can not simply be represented as $y=f(x)$, as both the variables are expressed in the form of a third variable. These types of form of functions are generally known as functions in parametric form. In exercise 5.6 of the chapter Continuity and Differentiability, we will learn about the derivatives of functions in parametric form. These concepts will help the students differentiate expressions involving variables $x$ and $y$ given in the form of a third variable or parameter. This article on the NCERT Solutions for Exercise 5.5 Class 12 Maths Chapter 5 - Continuity and Differentiability offers clear and step-by-step solutions for the problems given in the exercise, so that the students can develop their problem-solving ability and understand the logic behind these solutions. For syllabus, notes, and PDF, refer to this link: NCERT.
Answer:
Given equations are
$x = 2at^2, y = at^4$
Now, differentiate both w.r.t t
We get,
$\frac{dx}{dt}=\frac{d(2at^2)}{dt}= 4at$
Similarly,
$\frac{dy}{dt}=\frac{d(at^4)}{dt}= 4at^3$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{4at^3}{4at} = t^2$
Therefore, the answer is $\frac{dy}{dx}= t^2$
$x= a \cos \theta , y = b \cos \theta$
Answer:
Given equations are
$x= a \cos \theta , y = b \cos \theta$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(a\cos \theta)}{d\theta}= -a\sin \theta$
Similarly,
$\frac{dy}{d\theta}=\frac{d(b\cos \theta)}{d\theta}= -b\sin \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{-b\sin \theta}{-a\sin \theta} = \frac{b}{a}$
Therefore, answer is $\frac{dy}{dx}= \frac{b}{a}$
Answer:
Given equations are
$x = \sin t , y = \cos 2 t$
Now, differentiate both w.r.t t
We get,
$\frac{dx}{dt}=\frac{d(\sin t)}{dt}= \cos t$
Similarly,
$\frac{dy}{dt}=\frac{d(\cos 2t)}{dt}= -2\sin 2t = -4\sin t \cos t \ \ \ \ \ (\because \sin 2x = \sin x\cos x)$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{-4\sin t \cos t }{\cos t} = -4\sin t$
Therefore, the answer is $\frac{dy}{dx} = -4\sin t$
Answer:
Given equations are
$x = 4t , y = 4/t$
Now, differentiate both w.r.t t
We get,
$\frac{dx}{dt}=\frac{d(4 t)}{dt}= 4$
Similarly,
$\frac{dy}{dt}=\frac{d(\frac{4}{t})}{dt}= \frac{-4}{t^2}$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{ \frac{-4}{t^2} }{4} = \frac{-1}{t^2}$
Therefore, the answer is $\frac{dy}{dx} = \frac{-1}{t^2}$
Answer:
Given equations are
$x = \cos \theta - \cos 2\theta , y = \sin \theta - \sin 2 \theta$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(\cos \theta-\cos 2\theta)}{d\theta}= -\sin \theta -(-2\sin 2\theta) = 2\sin 2\theta - \sin \theta$
Similarly,
$\frac{dy}{d\theta}=\frac{d(\sin \theta - \sin 2\theta)}{d\theta}= \cos \theta -2\cos2 \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{\cos \theta -2\cos 2 \theta}{2\sin2\theta-\sin \theta}$
Therefore, answer is $\frac{dy}{dx}= \frac{\cos \theta -2\cos 2 \theta}{2\sin2\theta-\sin \theta}$
Answer:
Given equations are
$x = a ( \theta - \sin \theta ) , y = a ( 1+ \cos \theta )$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(a(\theta- \sin \theta))}{d\theta}= a(1-\cos \theta)$
Similarly,
$\frac{dy}{d\theta}=\frac{d(a(1+\cos \theta))}{d\theta}=-a\sin \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{-a\sin \theta}{a(1-\cos \theta)} = \frac{-\sin }{1-\cos \theta} =- \cot \frac{\theta}{2} \ \ \ \ \ \ \ (\cot \frac{x}{2}=\frac{\sin x}{1-\cos x})$
Therefore, the answer is $\frac{dy}{dx}=-\cot \frac{\theta}{2}$
Answer:
Given equations are
$x = \frac{\sin^3 t}{\sqrt{\cos 2t}}, \quad y = \frac{\cos^3 t}{\sqrt{\cos 2t}}$
Now, differentiate both w.r.t $t$:
$\frac{dx}{dt} = \frac{d\left( \frac{\sin^3 t}{\sqrt{\cos 2t}} \right)}{dt} = \frac{\sqrt{\cos 2t} \cdot \frac{d(\sin^3 t)}{dt} - \sin^3 t \cdot \frac{d(\sqrt{\cos 2t})}{dt}}{(\sqrt{\cos 2t})^2}$
$= \frac{3\sin^2 t \cos t \cdot \sqrt{\cos 2t} - \sin^3 t \cdot \frac{1}{2\sqrt{\cos 2t}} \cdot (-2\sin 2t)}{\cos 2t}$
$= \frac{3\sin^2 t \cos t \cdot \cos 2t + \sin^3 t \sin 2t}{\cos 2t \sqrt{\cos 2t}}$
$= \frac{\sin^3 t \sin 2t \left( 3 \cot t \cot 2t + 1 \right)}{\cos 2t \sqrt{\cos 2t}} \quad (\because \frac{\cos x}{\sin x} = \cot x)$
Similarly,
$\frac{dy}{dt} = \frac{d\left( \frac{\cos^3 t}{\sqrt{\cos 2t}} \right)}{dt} = \frac{\sqrt{\cos 2t} \cdot \frac{d(\cos^3 t)}{dt} - \cos^3 t \cdot \frac{d(\sqrt{\cos 2t})}{dt}}{(\sqrt{\cos 2t})^2}$
$= \frac{3\cos^2 t (-\sin t) \cdot \sqrt{\cos 2t} - \cos^3 t \cdot \frac{1}{2\sqrt{\cos 2t}} \cdot (-2\sin 2t)}{(\sqrt{\cos 2t})^2}$
$= \frac{-3\cos^2 t \sin t \cos 2t + \cos^3 t \sin 2t}{\cos 2t \sqrt{\cos 2t}}$
$= \frac{\sin 2t \cos^3 t \left( 1 - 3 \tan t \cot 2t \right)}{\cos 2t \sqrt{\cos 2t}}$
Now, $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\frac{\sin 2t \cos^3 t \left( 1 - 3 \tan t \cot 2t \right)}{\cos 2t \sqrt{\cos 2t}}}{\frac{\sin^3 t \sin 2t \left( 3 \cot t \cot 2t + 1 \right)}{\cos 2t \sqrt{\cos 2t}}}$
$= \frac{\cot^3 t \left( 1 - 3 \tan t \cot 2t \right)}{3 \cot t \cot 2t + 1}$
$= \frac{\cos^3 t \left( 1 - 3 \frac{\sin t}{\cos t} \frac{\cos 2t}{\sin 2t} \right)}{\sin^3 t \left( 3 \frac{\cos t}{\sin t} \frac{\cos 2t}{\sin 2t} + 1 \right)}$
$= \frac{\cos^2 t \left( \cos t \sin 2t - 3 \sin t \cos 2t \right)}{\sin^2 t \left( 3 \cos t \cos 2t + \sin t \sin 2t \right)}$
$= \frac{\cos^2 t \left( 2 \sin t \cos^2 t - 3 \sin t \cos^2 t + 3 \sin t \right)}{\sin^2 t \left( 3 \cos t - 6 \cos t \sin^2 t + 2 \sin^2 \cos t \right)}$
$= \frac{\sin t \cos t \left( -4 \cos^3 t + 3 \cos t \right)}{\sin t \cos t \left( 3 \sin t - 4 \sin^3 t \right)}$
$\frac{dy}{dx} = \frac{-4 \cos^3 t + 3 \cos t}{3 \sin t - 4 \sin^3 t} = \frac{-\cos 3t}{\sin 3t} = -\cot 3t \quad (\because \sin 3t = 3 \sin t - 4 \sin^3 t \text{ and } \cos 3t = 4 \cos^3 t - 3 \cos t)$
Therefore, the answer is $\frac{dy}{dx} = -\cot 3t$
Answer:
Given equations are
$x = a ( \cos t + \log \tan \frac{t}{2} ),y = a \sin t$
Now, differentiate both w.r.t t
We get,
$\frac{dx}{dt}=\frac{d(a ( \cos t + \log \tan \frac{t}{2} ))}{dt}= a(-\sin t + \frac{1}{\tan\frac{t}{2}}.\sec^2\frac{t}{2}.\frac{1}{2})$
$= a(-\sin t+\frac{1}{2}.\frac{\cos \frac{t}{2}}{\sin\frac{t}{2}}.\frac{1}{\cos^2\frac{t}{2}}) = a(-\sin t+\frac{1}{2\sin \frac{t}{2}\cos \frac{t}{2}})$
$=a(-\sin t+\frac{1}{\sin 2.\frac{t}{2}} ) = a(\frac{-\sin^2t+1}{\sin t})= a(\frac{\cos^2t}{\sin t})$
Similarly,
$\frac{dy}{dt}=\frac{d(a\sin t)}{dt}= a\cos t$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{a \cos t }{ a(\frac{\cos^2t}{\sin t})} = \frac{\sin t}{\cos t} = \tan t$
Therefore, the answer is $\frac{dy}{dx} = \tan t$
Answer:
Given equations are
$x = a \sec \theta , y = b \ tan \theta$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(a\sec \theta)}{d\theta}= a\sec \theta \tan \theta$
Similarly,
$\frac{dy}{d\theta}=\frac{d(b\tan \theta)}{d\theta}= b\sec^2 \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{b\sec^2 \theta}{a\sec\theta\tan \theta} = \frac{b\sec\theta}{a\tan \theta}= \frac{b\frac{1}{\cos\theta}}{a\frac{\sin \theta}{\cos \theta}} = \frac{b }{a\sin \theta} = \frac{b cosec \theta}{a}$
Therefore, the answer is $\frac{dy}{dx} = \frac{b cosec \theta}{a}$
Answer:
Given equations are
$x = a ( \cos \theta + \theta \sin \theta ) , y = a ( \sin \theta - \theta \cos \theta )$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(a(\cos \theta+ \theta\sin \theta))}{d\theta}= a(-\sin \theta+\sin \theta+ \theta\cos \theta)= a \theta\cos \theta$
Similarly,
$\frac{dy}{d\theta}=\frac{d(a(\sin \theta- \theta\cos \theta))}{d\theta}= a(\cos \theta-\cos \theta+ \theta\sin \theta) = a \theta\sin \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{a \theta\sin \theta}{a \theta\cos \theta} = \tan \theta$
Therefore, the answer is $\frac{dy}{dx}= \tan \theta$
Answer:
Given equations are
$x = \sqrt {a ^{\sin ^{-1}t}} , y = \sqrt { a ^{ \cos ^{-1}t}}$
$xy=\sqrt{a^{sin^{-1}t+cos^{-1}t}}\\since\ sin^{-1}x+cos^{-1}x=\frac{\pi}{2}\\xy=a^{\frac{\pi}{2}}=constant=c$
differentiating with respect to x
$x\frac{dy}{dx}+y=0\\\frac{dy}{dx}=\frac{-y}{x}$
Also Read,
The main topics covered in Chapter 5 of continuity and differentiability, exercises 5.6 are:
Also, read,
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Frequently Asked Questions (FAQs)
Given x = sin (t)
dx/dt = cos(t)
y=sin(t)
As y is not dependent on the x.
dy/dx = 0
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