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

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Answer 1 · 2015-12-16T00:52:17

$f''(x)=3xe^(x^3)(3x^3+2)$

The first derivative (using the chain rule):

$f'(x)=d/dx[e^(x^3)]=e^(x^3)d/dx[x^3]=3x^2e^(x^3)$

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

$f''(x)=3x^2d/dx[e^(x^3)]+e^(x^3)d/dx[3x^2]$

$f''(x)=3x^2(3x^2e^(x^3))+e^(x^3)(6x)$

$f''(x)=3xe^(x^3)(3x^3+2)$

What is the second derivative of f(x)= e^(x^3)f(x)=ex3f(x)= e^(x^3)?