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Differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiation2 Marks
Question
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{a}(1-\cos\theta)\text{ and y}=\text{a}(\theta+\sin\theta)\text{ at }\theta=\frac{\pi}{1}$
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Answer

We have, $\text{x}=\text{a}(1-\cos\theta)\text{ and y}=\text{a}(\theta+\sin\theta)$
$\therefore\frac{\text{dx}}{\text{d}\theta}=\frac{\text{d}}{\text{d}\theta}[\text{a}(1-\cos\theta)]=\text{a}(\sin\theta)$
and
$\frac{\text{dx}}{\text{d}\theta}=\frac{\text{d}}{\text{d}\theta}[\text{a}(\theta+\sin\theta)]=\text{a}(1+\cos\theta)$
$\therefore\Big[\frac{\text{dy}}{\text{dx}}\Big]_{\theta=\frac{\pi}{2}}=\bigg[\frac{\frac{\text{dy}}{\text{d}\theta}}{\frac{\text{dx}}{\text{d}\theta}}\bigg]_{\theta=\frac{\pi}{2}} \\ =\Big[\frac{\text{a}(1+\cos\theta)}{\text{a}(\sin\theta)}\Big]_{\theta=\frac{\pi}{2}}=\frac{\text{a}(1+0)}{\text{a}}=1$

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