Question

Differentiate the following functions w.r.t.$x$
${\left(a^{\sqrt{x}}\right)}^{\sin x}$

Answer 1

$y={\left(a^{\sqrt{x}}\right)}^{\sin x}$

$\Rightarrow y=a^{\sqrt{x}\sin x}$ where ${a^m}^n=a^{mn}$

Let $u=\sqrt{x}\sin x$ and $y=a^u$

then $u=\sqrt{x}\sin x$

$\Rightarrow \frac{du}{dx}=\sqrt{x}\frac{d}{dx}\left(\sin x\right)+\sin x\frac{d}{dx}\left(\sqrt{x}\right)$ using product rule of differentiation

$=\sqrt{x}\cos x+\frac{\sin x}{2\sqrt{x}}$

and $y=a^u\Rightarrow \frac{dy}{du}=a^u\log a$

$\therefore \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$ by substituting the above values

$=a^u\log a\times (\sqrt{x}\cos x+\frac{\sin x}{2\sqrt{x}})$

$=a^{\sqrt{x}\sin x}\log a(\sqrt{x}\cos x+\frac{\sin x}{2\sqrt{x}})$