Derivative of e^sin^2 x with respect to cos x is:

The correct answer is Option (3) → $-2 \cos x e^{\sin^2 x}$ ##

Let $u = e^{\sin^2 x}$ and $v = \cos x$.

$\frac{du}{dx} = e^{\sin^2 x} (2 \sin x \cos x)$

and $\frac{dv}{dx} = -\sin x$

$\text{Thus, } \frac{du}{dv} = \frac{du/dx}{dv/dx}$

$= \frac{e^{\sin^2 x} \cdot 2 \sin x \cos x}{-\sin x}$

$= -2 \cos x e^{\sin^2 x}$