Efan Halliday

2021-09-20

Find the derivatives of the functions
$y={\left(5-2x\right)}^{-3}+\frac{1}{8}{\left(\frac{2}{x}+1\right)}^{4}$

Ezra Herbert

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

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