Question

Differential the following functions w.r.t. $x$:
${\left(\frac{4x-1}{5x+4}\right)}^{1/3}$

Answer 1

$y=(\frac{4x-1}{5x+4}{)}^{\frac{1}{3}}$ chain Rule

$\frac{dy}{dx}=\frac{1}{3}\left(\frac{4x-1}{5x+4}{)}^{-\frac{2}{3}}\right[\frac{(5x+4)\left(4\right)-(4x-1)\left(5\right)}{(5x+4{)}^2}]$

$=\frac{1}{3}\left(\frac{5x+4}{4x-1}{)}^{\frac{2}{3}}\right[\frac{21}{(5x+4{)}^2}]$

$=\frac{7}{(4x-1{)}^{\frac{2}{3}}(5x+4{)}^{\frac{4}{3}}}$