nicekikah

2021-10-15

Derivatives Find and simplify the derivative of the following functions
$h\left(x\right)=\left(5{x}^{7}+5x\right)\left(6{x}^{3}+3{x}^{2}+3\right)$

Brighton

Step 1
Given,
$h\left(x\right)=\left(5{x}^{7}+5x\right)\left(6{x}^{3}+3{x}^{2}+3\right)$
Step 2
Concept used:
$\frac{d}{dx}\left(f\left(x\right)g\left(x\right)\right)=g\left(x\right)\frac{d}{dx}\left(f\left(x\right)\right)+f\left(x\right)\frac{d}{dx}\left(g\left(x\right)\right)$
$\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$
$\frac{d}{dx}\left(c\right)=0$
Step 3
On differentiating with respect to x, to obtain
$\frac{d}{dx}\left(h\left(x\right)\right)=\frac{d}{dx}\left[\left(5{x}^{7}+5x\right)\left(6{x}^{3}+3{x}^{2}+3\right)\right]$
${h}^{\prime }\left(x\right)=\left(6{x}^{3}+3{x}^{2}+3\right)\frac{d}{dx}\left(5{x}^{7}+5x\right)+\left(5{x}^{7}+5x\right)\frac{d}{dx}\left(6{x}^{3}+3{x}^{2}+3\right)$
${h}^{\prime }\left(x\right)=\left(6{x}^{3}+3{x}^{2}+3\right)\left(35{x}^{6}+5\right)+\left(5{x}^{7}+5x\right)\left(18{x}^{2}+6x+0\right)$
${h}^{\prime }\left(x\right)=210{x}^{9}+30{x}^{3}+105{x}^{8}+15{x}^{2}+105{x}^{6}+15+90{x}^{9}+30{x}^{8}+90{x}^{3}+30{x}^{2}$
${h}^{\prime }\left(x\right)=300{x}^{9}+135{x}^{8}+105{x}^{6}+120{x}^{3}+45{x}^{2}+15$

Jeffrey Jordon