illusiia

2021-10-05

Derivatives Find and simplify the derivative of the following functions
$p\left(x\right)=\frac{4{x}^{3}+3x+1}{2{x}^{5}}$

Step 1
Given: $p\left(x\right)=\frac{4{x}^{3}+3x+1}{2{x}^{5}}$
for finding derivative of given function, first we simplify given expression then find derivative of it
Step 2
so,
$p\left(x\right)=\frac{4{x}^{3}+3x+1}{2{x}^{5}}$
$=\frac{4{x}^{3}}{2{x}^{5}}+\frac{2x}{2{x}^{5}}+\frac{1}{2{x}^{5}}$
$=2{x}^{-2}+{x}^{-4}+\frac{{x}^{-5}}{2}$
now differentiating it
so,
$\frac{d}{dx}\left[p\left(x\right)\right]=\frac{d}{dx}\left[2{x}^{-2}+{x}^{-4}+\frac{{x}^{-5}}{2}\right]$
$=2\frac{d}{dx}\left({x}^{-2}\right)+\frac{d}{dx}\left({x}^{-4}\right)+\frac{1}{2}\frac{d}{dx}\left({x}^{-5}\right)$
$\left(\because \frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}\right)$
$=2\left(-2{x}^{-3}\right)+\left(-4{x}^{-5}\right)+\frac{1}{2}\left(-5{x}^{-6}\right)$
$=-\frac{4}{{x}^{3}}-\frac{4}{{x}^{5}}-\frac{5}{2{x}^{6}}$
$=-\frac{8{x}^{3}+8x+5}{2{x}^{6}}$
Step 3
hence, derivative of given function is $-\frac{8{x}^{3}+8x+5}{2{x}^{6}}$.

Jeffrey Jordon