Question

How do you use the Product Rule to find the derivative of `f(x) = (x^2 + 2x +3)e^-x`?

Answer 1

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

Explanation:

Notice that you can write your function as

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

where `g(x) = (x^2 + 2x + 3)` and `h(x) = e^(-x)`. The product rule allows you to differentiate functions that can be written as the the product of two other functions by using the formula

`color(blue)(d/dx(f(x)) = [d/dx(g(x))] * h(x) + g(x) * d/dx(h(x)))`

If you take into account the fact that

`d/dx(e^(-x)) = -e^(-x)`

you can write

`d/dx(f(x)) = [d/dx(x^2 + 2x + 3)] * e^(-x) + (x^2 + 2x + 3) * d/dx(e^(-x))`

`f^' = (2x + 2) * e^(-x) + (x^2 + 2x + 3) * (-e^(-x))`

`f^' = e^(-x) * (color(red)(cancel(color(black)(2x))) + 2 - x^2 - color(red)(cancel(color(black)(2x))) - 3)`

`f^' = e^(-x) * (-x^2 -1)`

`f^' = color(green)(-e^(-x) * (x^2 + 1))`