Find the derivative of f where f(x)=a^x.

This is a difficult question that you only need to know the result of.However, it's a good exercise to derive it. 

Starting with f(x)=a^x we can take the natural logarithm of both sides (so we can use one of its properties).

This gives us ln(f(x))=ln(a^x), from the natural logarithms properties we know this is equal to ln(f(x))=x*ln(a).

Now using the chain rule we can differentiate both sides,

d(ln(f(x)))/dx= f'(x)/f(x), d(x*ln(a))/dx=ln(a)

so we now have f'(x)/f(x)=ln(a). Recalling that f(x)=a^x this gives us the answer,

f'(x)=a^xln(a).