Question

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

Answer 1

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

Explanation:

differentiate using the `color(blue)"product rule"`

`"Given "f(x)=g(x).h(x)" then"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=g(x)h'(x)+h(x)g'(x))color(white)(2/2)|)))`

`"here "g(x)=x^2rArrg'(x)=2x`

`"and "h(x)=e^-xrArrh'(x)=e^-x.d/dx(-x)=-e^-x`

`rArrf'(x)=x^2(-e^-x)+e^-x(2x)`

`color(white)(rArrf'(x))=2xe^-x-x^2e^-x`