Question

Derivative of y= 2^(sin pi x)

Answer 1

Take log of both sides
$\ln y = \ln(2^{\sin \pi x})$
Use log property to get "x" out of the exponent
$\ln y = (\ln 2) (\sin \pi x)$
Differentiate
$\frac{dy}{y} = (\pi \ln 2)(cos \pi x) dx$
Move y to Right side  (Note y = original expression)
$\frac{dy}{dx} = (\pi \ln 2) (\cos \pi x) 2^{\sin \pi x}$

In general, whenever you have a function in the form
$y = a^{f(x)}$
The derivative will be
$\frac{dy}{dx} = (\ln a) (f'(x))a^{f(x)}$