Derivative of f(x)=(1+x^2)e^(-x^2)?

Question

What is the derivative of:

f(x)=(1+x^2)e^(-x^2)


i got f'(x)=2x(1+x^2)e^(-2x) , but i think thats wrong...

Answer 1

Yes that is wrong, but at least you tried.

The first rule to apply here is the multiplicative rule:

if f(x) = g(x)h(x) then f'(x) = g'(x)h(x) + g(x)h'(x)

In this case:
g(x) = 1+x^2
h(x) = e^(-x^2)

g'(x) is easy to compute so let's move on to h'(x). Here we need the chain rule.

if h(x) = j(k(x)) then dh/dx = (dj/du)(dk/dx)

Here:

u = k(x) = -(x^2)
j(u) = e^u

When you use the chain rule, you have to substitute k(x) for u in the derivative. For example, in this case

dj/du = j'(u) = e^u = e^(-x^2)

so h'(x) = e^(-x^2)k'(x)

Since it seems you know how to differentiate quadratics, you now have all the pieces you need to compute h'(x), g'(x) and finally f'(x).