Question

How do you find the derivative of `f(x)=xe^(-(x^2)/2)`

Answer 1

You can use the Product Rule where:

if `f(x)=g(x)*h(x)`

`f'(x)=g'(x)h(x)+g(x)h'(x)`

in your case `g(x)=x` but `h(x)=e^(-x^2/2)` which needs the Chain Rule to be derived (here you derive the `e` as it is and multiply times the derivative of the exponent).

So you get:
`g(x)=x`
`g'(x)=1`
`h(x)=e^(-x^2/2)`
`h'(x)=e^(-x^2/2)(-2x/2)`

and finally your derivative:

`f'(x)=1e^(-x^2/2)+xe^(-x^2/2)(-2x/2)=`
`=e^(-x^2/2)(1-x^2)`