Cheyanne Leigh

2021-09-30

Differentiate the function.
$G\left(x\right)=\sqrt[5]{{x}^{6}+6x}$
${G}^{\prime }\left(x\right)=$

curwyrm

Step 1
We have to differentiate the given function:
$G\left(x\right)=\sqrt[5]{{x}^{6}+6x}$
Rewriting the given function,
$G\left(x\right)=\sqrt[5]{{x}^{6}+6x}$
$={\left({x}^{6}+6x\right)}^{\frac{1}{5}}$
We know the formula of derivatives,
$\frac{{dx}^{n}}{dx}=n{x}^{n-1}$
$\frac{d{\left(f\left(x\right)\right)}^{n}}{dx}=n{\left(f\left(x\right)\right)}^{n-1}\frac{df\left(x\right)}{dx}$
Step 2
Applying above formula for the given function, we get
$\frac{dG\left(x\right)}{dx}=\frac{d{\left({x}^{6}+6x\right)}^{\frac{1}{5}}}{dx}$
${G}^{\prime }\left(x\right)=\frac{1}{5}{\left({x}^{6}+6x\right)}^{\frac{1}{5}-1}\frac{d\left({x}^{6}+6x\right)}{dx}$
$=\frac{1}{5}{\left({x}^{6}+6x\right)}^{\frac{1-5}{5}}\left(6{x}^{6-1}+6\frac{dx}{dx}\right)$
$=\frac{1}{5}{\left({x}^{6}+6x\right)}^{\frac{-4}{5}}\left(6{x}^{5}+6\right)$
$=\frac{\left(6{x}^{5}+6\right)}{5{\left({x}^{6}+6x\right)}^{\frac{4}{5}}}$
Hence,
${G}^{\prime }\left(x\right)=\frac{\left(6{x}^{5}+6\right)}{5{\left({x}^{6}+6x\right)}^{\frac{4}{5}}}$.

Jeffrey Jordon