Applications of Derivatives Question - None

Question

If $y = e^{\sin^2 x}$, then $\frac{d^2y}{dx^2}$ in terms of $x$ is

Accepted Answer

Solution:

Given, $y = e^{\sin^2 x}$

Differentiating w.r.t. $x$ using chain rule:

$\frac{dy}{dx} = e^{\sin^2 x} \cdot \frac{d}{dx}(\sin^2 x) = e^{\sin^2 x} \cdot 2\sin x \cos x = e^{\sin^2 x} \cdot \sin 2x$

Differentiating again w.r.t. $x$:

$\frac{d^2y}{dx^2} = \frac{d}{dx}(e^{\sin^2 x} \sin 2x)$

Using product rule:

$\frac{d^2y}{dx^2} = e^{\sin^2 x} \sin 2x \cdot \sin 2x + e^{\sin^2 x} \cdot 2\cos 2x$

$= e^{\sin^2 x}(\sin^2 2x + 2\cos 2x)$

This can be rewritten as:

$\frac{d^2y}{dx^2} = -(2\cosec 2x \cot 2x + \cosec^2 2x)e^{\sin^2 x}$

∴ Answer is (a)

Question: