Derivative of exponential function

Question

1) $f(t) = (\ln 5)^t$

what is the $f'(t)$? I tried $t\ln(5)$ but it was wrong.

2) $f(x) = x^{\Large π^6} + (π^4)^x$

This one I did not attempt in it because I find it confusing little bit.

Answer 1

For no. $1$, let $y=f(t)$, then \begin{align} y&=(\ln5)^t\\ \ln y&=t\ln(\ln5)\qquad;\qquad\text{take $\ln$ both of sides} \\ y&=e^{\large t\ln(\ln5)}\qquad;\qquad \ln a=b\;\Rightarrow\;a=e^b\\ \frac{dy}{dt}&=\frac{d}{dt}\left[e^{\large t\ln(\ln5)}\right]\qquad;\qquad \text{derive both of sides and let $u=t\ln(\ln5)$}\\ y'&=\frac{d}{du}\left[e^{\large u}\right]\cdot\frac{du}{dt}\qquad;\qquad\text{apply chain rule}\\ f'(t)&=e^{\large u}\cdot\ln(\ln5)\\ &=e^{\large t\ln(\ln5)}\cdot\ln(\ln5)\\ &=(\ln5)^t\cdot\ln(\ln5).\\ \end{align} Similar with $1$, for $2$, let $y=\pi^{4x}$, then $y=e^{4x\ln\pi}$. Hence $$ f'(x)=\pi^6 x^{\Large \pi^6-1}+(4\ln\pi)\pi^{4x}. $$ Remark : $$ y=a^{\large f(x)}\qquad\Rightarrow\qquad y'=a^{\large f(x)}\cdot f'(x),\qquad;\qquad\text{for $a$ is a constant.} $$