Let [tex]b>0[/tex] be any real number with [tex]b\neq e[/tex]. Then you can write
[tex]b^x=e^{\ln b^x}=e^{x\ln b}[/tex]
Now, by the chain rule for differentiation, the derivative is
[tex]\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[/tex]
