Find the derivatives of the functions y=(5−2x)−3+18(2x+1)4

Step 1
Here given
${\displaystyle y={(5-2x)}^{-3}+\frac{1}{8}{(\frac{2}{x}+1)}^4}$
Step 2
Now, differentiate the given function x with respect to x
${\displaystyle \frac{dy}{dx}=\frac{d}{dx}\{{(5-2x)}^3+\frac{1}{8}{(\frac{2}{x}+1)}^4\}}$
${\displaystyle \Rightarrow \frac{dy}{dx}=\frac{d}{dx}{(5-2x)}^{-3}+\frac{1}{8}\frac{d}{dx}{(\frac{2}{x}+1)}^4}$
${\displaystyle \Rightarrow \frac{dy}{dx}=-3{(5-2x)}^{-4}\frac{d}{dx}(5-2x)+\frac{4}{8}{(\frac{2}{x}+1)}^3\frac{d}{dx}(\frac{2}{x}+1)}$
${\displaystyle \Rightarrow \frac{dy}{dx}=-3{(5-2x)}^{-4}(-2)+\frac{4}{8}{(\frac{2}{x}+1)}^3\{2\frac{d}{dx}\left(\frac{1}{x}\right)+\frac{d}{dx}\left(1\right)\}}$
${\displaystyle \Rightarrow \frac{dy}{dx}=6{(5-2x)}^{-4}+\frac{1}{2}{(\frac{2}{x}+1)}^3\left(\frac{2}{x^2}\right)}$
${\displaystyle \Rightarrow \frac{dy}{dx}=6{(5-2x)}^4+\left(\frac{1}{x^2}\right){(\frac{2}{x}+1)}^3}$