Question

Find the derivative of the following functions (it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $(ax+b){}^n(cx+d){}^m$

Answer 1

Let $f\left(x\right)=(ax+b){}^n(cx+d){}^m$
Thus using product rule,
$f^{\prime }\left(x\right)={(ax+b)}^2\left\{mc{(cx+d)}^{m-1}\right\}+{(cx+d)}^m\left\{na{(ax+b)}^{n-1}\right\}$
$={(ax+b)}^{n-1}{(cx+d)}^{m-1}[mc(ax+b)+na(cx+d)]$