Question

What is the derivative of `e^((x^2)/2)`?

Answer 1

Use the chain rule to find that

`d/dx e^(x^2/2) = xe^(x^2/2)`

Explanation:

The chain rule states that given two differentiable functions `f` and `g`

`d/dxf@g(x) = f'(g(x))*g'(x)`

In this case, let `f(x) = e^x` and `g(x) = 1/2x^2`

Then `f'(x) = e^x` and `g'(x) = 1/2(2x) = x`

So, by the chain rule, we have

`d/dxe^(x^2/2) = d/dxf@g(x)`

`=f'(g(x))*g'(x)`

`=e^(g(x))*x`

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