Take log of both sides
[tex]\ln y = \ln(2^{\sin \pi x})[/tex]
Use log property to get "x" out of the exponent
[tex]\ln y = (\ln 2) (\sin \pi x)[/tex]
Differentiate
[tex]\frac{dy}{y} = (\pi \ln 2)(cos \pi x) dx[/tex]
Move y to Right side (Note y = original expression)
[tex]\frac{dy}{dx} = (\pi \ln 2) (\cos \pi x) 2^{\sin \pi x}[/tex]
In general, whenever you have a function in the form
[tex]y = a^{f(x)}[/tex]
The derivative will be
[tex]\frac{dy}{dx} = (\ln a) (f'(x))a^{f(x)}[/tex]
