Question

Differentiate w.r.to $x$

$y=2^{3^{\sin x^2}}$

Answer 1

$y=2^{3\sin x^2}$

we know that if $y=a^{f\left(x\right)}$ then $\frac{dy}{dx}=\left(\ln a\right)f^{\prime }\left(x\right)\left(a^{f\left(x\right)}\right)$

$\therefore \frac{dy}{dx}=\left(\ln 2\right)\left(2^{3\sin x^2}\right)\left(\frac{d\left(3\sin x^2\right)}{dx}\right)$

$=\left(\ln 2\right)\left(\ln 3\right)\left(2^{3\sin x^2}\right)\left(3^{\sin x^2}\right)\left(2x\right)\left(\cos x^2\right)$

$\therefore \frac{dy}{dx}=\left(2\ln 2\ln 3\right)\left(x{\cos }^{2}x\right)\left(2^{3\sin x^2}\right)\left(3^{\sin x^2}\right)$