Let \(b>0\) be any real number with \(b
eq e\). Then you can write
\(b^x=e^{\ln b^x}=e^{x\ln b}\)
Now, by the chain rule for differentiation, the derivative is
\(\dfrac{\mathrm d}{\mathrm dx}e^{x\ln b}=e^{x\ln b}\dfrac{\mathrm d}{\mathrm dx}(x\ln b)=\ln b~e^{x\ln b}=\ln b~b^x\)
