Question

What is the second derivative of `f(x)= xe^(x^2)`?

Answer 1

`f''(x) = (4x^3+6x)e^(x^2)`

Explanation:

`f(x) = xe^(x^2)`

Apply product rule..

`f'(x) = x * d/dx(e^(x^2)) + d/dx x * e^(x^2)`

Apply standard differential and chain rule.

`f'(x) = x* e^(x^2)* 2x + 1 * e^(x^2)`

`= (2x^2+1)e^(x^2)`

Apply product rule.

`f''(x) = (2x^2+1) * d/dx (e^(x^2)) + d/dx (2x^2+1) * e^(x^2)`

`= (2x^2+1) * 2x* e^(x^2) + 4x*e^(x^2)`

`= (4x^3 +2x + 4x)e^(x^2)`

`= (4x^3+6x)e^(x^2)`