Differentiate ^2 x w.r.t. e^ x . - Vidyadip

Question

Differentiate $\sin^2 x\ \text{w.r.t}. e^{\cos x}$.

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Answer

Let $u (x) = \sin^2 x$ and $v (x) = e^{\cos x}$.
We want to find $\frac{d u}{d v}=\frac{d u / d x}{d v / d x}$.
Clearly $\frac{d u}{d x} = 2 \sin x \cos x$ and $\frac{d v}{d x} = e^{\cos x} (– \sin x) = – (\sin x) e^{\cos x}$
$\therefore \frac{d u}{d v}=\frac{2 \sin x \cos x}{-\sin x e^{\cos x}}=-\frac{2 \cos x}{e^{\cos x}}$

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