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

1 Answer
Aug 9, 2015

Answer:

#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))#