Question

How do you find the derivative and double derivative of the equation `f(x)=xe^(-x^2)`?

Answer 1

`dy/dx=-2x^2e^(-x^2)+e^(-x^2)`
`(d^2y)/(dx^2)=2xe^(-x^2)(2x^2-3x)`

Explanation:

Given -

`y=xe^(-x^2)`

`dy/dx=x.e^(-x^2)(-2x)+e^(-x^2)(1)`

`dy/dx=-2x^2e^(-x^2)+e^(-x^2)`

`(d^2y)/(dx^2)=[-2x^2.e^(-x^2)(-2x)+e^(-x^2)(-4x)]+[e^(-x^2)(-2x)]`

`(d^2y)/(dx^2)=4x^3e^(-x^2)-4xe^(-x^2)-2xe^(-x^2)`

`(d^2y)/(dx^2)=4x^3e^(-x^2)-6xe^(-x^2)`

`(d^2y)/(dx^2)=2xe^(-x^2)(2x^2-3x)`