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  • Provide solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Very short answers question 14

Provide solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Very short answers question 14

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Answer:

\text { The value of } \frac{d y}{d x}=-\tan \theta / 2

Given:

\text { If } x=a(\theta+\sin \theta), y=a(1+\cos \theta) \text { find } \frac{d y}{d x}

Hint:

\begin{aligned} &\frac{d}{d x}(\sin x)=\cos x \\\\ &\frac{d}{d x}(\cos x)=-\sin x \end{aligned}

Solution:  

\begin{aligned} &\frac{d x}{d \theta}=a \frac{d}{d \theta}(\theta+\sin \theta) \\\\ &=a[1+\cos \theta] \end{aligned}

\begin{aligned} &\frac{d y}{d \theta}=a \frac{d}{d \theta}[1+\cos \theta] \\\\ &=-a \sin \theta \end{aligned}

\begin{aligned} &\frac{d y}{d x}=\frac{-a \sin \theta}{a(1+\cos \theta)} \\\\ &=\frac{-\sin \theta}{1+\cos \theta} \end{aligned}

=\frac{-2 \sin ^{\theta} / 2 \cos ^{\theta} / 2}{2 \cos ^{2} \theta / 2}        using formula of half angle

=-\tan \theta / 2

So, the answer of  \frac{d y}{d x}=-\tan \theta / 2

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