2021-10-04

Derivatives Find and simplify the derivative of the following functions
$y=\frac{{w}^{4}+5{w}^{2}+w}{{w}^{2}}$

grbavit

Step 1
Given: $y=\frac{{w}^{4}+5{w}^{2}+w}{{w}^{2}}$
Step 2
Explanation:
$y=\frac{{w}^{4}+5{w}^{2}+w}{{w}^{2}}$
We can re-write it as
$y=\frac{{w}^{4}}{{w}^{2}}+\frac{5{w}^{2}}{{w}^{2}}+\frac{w}{{w}^{2}}$
$y={w}^{2}+5+\frac{1}{w}$
$y={w}^{2}+5+{w}^{-1}$
Differentiate both sides with respect to w , we get
$\frac{dy}{dw}=\frac{d}{dw}\left({w}^{2}+5+{w}^{-1}\right)$
$\frac{dy}{dw}=\frac{d}{dw}\left({w}^{2}\right)+\frac{d}{dw}\left(5\right)+\frac{d}{dw}\left({w}^{-1}\right)$

$\frac{dy}{dw}=2w-{w}^{-2}$
$\frac{dy}{dw}=2w-\frac{1}{{w}^{2}}$
$\frac{dy}{dw}=\frac{2{w}^{3}-1}{{w}^{2}}$

Jeffrey Jordon