# How do you find (d^2y)/(dx^2) given y=root3(x)?

Oct 31, 2016

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - \frac{2}{9 {x}^{\frac{5}{3}}}$

#### Explanation:

The notation $\frac{{d}^{2} y}{\mathrm{dx}} ^ 2$ means to find the 2nd derivative.

So, we differentiate once, and we differentiate that result again.

$y = \sqrt[3]{x} \to y = {x}^{\frac{1}{3}}$

$y ' = \frac{1}{3} {x}^{- \frac{2}{3}}$

The second derivative is found by differentiating the first derivative.

$y ' ' = \frac{1}{3} \left(- \frac{2}{3}\right) {x}^{- \frac{5}{3}} = - \frac{2}{9} {x}^{- \frac{5}{3}} = - \frac{2}{9 {x}^{\frac{5}{3}}}$

Hopefully this helps!