Question

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

Answer 1

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

Explanation:

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)`