The integral of a^u where u is a function of x is given as

1/ln(a) a^u + C

but if I take the derivative of that using the product rule we have C=0 because it's a constant and the derivative of 1/ln(a) = 0

so f'g=0 and for fg' we have

ua^(u-1)/ln(a) which does not look equal to a^u (but maybe it is).

I'm assuming u is a function of x but I'm setting it as u = x so that u' =1

taking the derivative of the integral should give back the original integral function correct? what am I doing wrong?

thanks!