Derivative of $\sin ^2 x$ with respect to $\mathrm{e}^{\cos x}$ is

Derivative of $\sin ^2 x$ with respect to $\mathrm{e}^{\cos x}$ is
  1. $2 \sin x \cos ^2 x \mathrm{e}^{\cos x}$
  2. $\frac{2 \cos x}{\mathrm{e}^{\cos x}}$
  3. $\frac{2 \sin x}{\mathrm{e}^{\cos x}}$
  4. $\frac{-2 \cos x}{e^{\cos x}}$

Solution

Let $\mathrm{u}=\sin ^2 x, \mathrm{v}=\mathrm{e}^{\cos x}$ $\mathrm{u}=\sin ^2 x$ Differentiating w.r.t. $x$, we get $\frac{\mathrm{du}}{\mathrm{~d} x}=2 \sin x \cdot \cos x$ Consider, $\mathrm{v}=\mathrm{e}^{\cos x}$ Differentiating w.r.t. $x$, we get $\begin{aligned} & \therefore \quad \frac{\mathrm{dv}}{\mathrm{~d} x}=-\mathrm{e}^{\cos x} \cdot \sin x \\ & \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 \sin x \cdot \cos x}{-e^{\cos x} \cdot \sin x} \\ & =\frac{-2 \cos x}{\mathrm{e}^{\cos x}} \end{aligned}$

Asked in: MHT CET 2024 (03 May Shift 2)

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