Find differentiation of (1+ ) w.r.t. ^ -1 .

Question

Find differentiation of $\log (1+\theta)$ w.r.t. $\sin ^{-1} \theta$.

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Answer

Suppose $\quad y=\log (1+\theta)$ and $x=\sin ^{-1} \theta$
$\therefore \quad \frac{d y}{d \theta}=\frac{1}{1+\theta}$
and $\quad \frac{d x}{d \theta}=\frac{1}{\sqrt{1-\theta^2}}$
$\therefore \quad \frac{d y}{d x}=\frac{d y / d \theta}{d x / d \theta}=\frac{1 / 1+\theta}{\frac{1}{\sqrt{1-\theta^2}}}$
$
=\sqrt{\frac{1-\theta}{1+\theta}}
$

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