Question

what is f'(1) if f(x)=tan(3^x)

Answer 1

\[f(x)=\tan(3^x)\]
Take the derivative using the chain rule:
\[f'(x)=\frac{d}{dx}\tan(3^x)=\sec^2(3^x)\frac{d}{dx}(3^x)\]
If you're like me, you don't remember the derivative rule for exponential functions not in base \(e\). Luckily for us, you can easily derive it. Consider \(b^x\) with \(b>0\), then
\[\large b^x=e^{\ln b^x}=e^{x\ln b}~~\Rightarrow~~\frac{d}{dx}e^{x\ln b}=\ln b~e^{x\ln b}=\ln b~e^{\ln b^x}=\ln b(b^x)\]
So you have
\[f'(x)=\ln3\sec^2(3^x)3^x\]
Plug in \(x=1\) and you're done.