What is the second derivative of f(x)= e^(x^3)?

Dec 16, 2015

$f ' ' \left(x\right) = 3 x {e}^{{x}^{3}} \left(3 {x}^{3} + 2\right)$

Explanation:

The first derivative (using the chain rule):

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left[{e}^{{x}^{3}}\right] = {e}^{{x}^{3}} \frac{d}{\mathrm{dx}} \left[{x}^{3}\right] = 3 {x}^{2} {e}^{{x}^{3}}$

Now, use the product rule as well to find the second derivative:

$f ' ' \left(x\right) = 3 {x}^{2} \frac{d}{\mathrm{dx}} \left[{e}^{{x}^{3}}\right] + {e}^{{x}^{3}} \frac{d}{\mathrm{dx}} \left[3 {x}^{2}\right]$

$f ' ' \left(x\right) = 3 {x}^{2} \left(3 {x}^{2} {e}^{{x}^{3}}\right) + {e}^{{x}^{3}} \left(6 x\right)$

$f ' ' \left(x\right) = 3 x {e}^{{x}^{3}} \left(3 {x}^{3} + 2\right)$