Question

how do I find the derivative of a fraction raised to a power

Answer 1

Let $b>0$ be any real number with $b\neq e$. Then you can write

$b^x=e^{\ln b^x}=e^{x\ln b}$

Now, by the chain rule for differentiation, the derivative is

$\dfrac{\mathrm d}{\mathrm dx}e^{x\ln b}=e^{x\ln b}\dfrac{\mathrm d}{\mathrm dx}(x\ln b)=\ln b~e^{x\ln b}=\ln b~b^x$