Question

find $g(x)$ for $g(x)=7x^{2}+7x^{\frac{1}{5}}$.

Explanation:

Step1: Find the first - derivative

Use the power rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$. For $g(x)=7x^2+7x^{\frac{1}{5}}$, $g'(x)=\frac{d}{dx}(7x^2)+\frac{d}{dx}(7x^{\frac{1}{5}})=2\times7x^{2 - 1}+\frac{1}{5}\times7x^{\frac{1}{5}-1}=14x+\frac{7}{5}x^{-\frac{4}{5}}$.

Step2: Find the second - derivative

Differentiate $g'(x)$ again using the power rule. $\frac{d}{dx}(14x)=14$ and $\frac{d}{dx}(\frac{7}{5}x^{-\frac{4}{5}})=-\frac{4}{5}\times\frac{7}{5}x^{-\frac{4}{5}-1}=-\frac{28}{25}x^{-\frac{9}{5}}$. So $g''(x)=14-\frac{28}{25x^{\frac{9}{5}}}$.

Answer:

$g''(x)=14-\frac{28}{25x^{\frac{9}{5}}}$