How do you find the first and second derivatives of #y = (x^2 + 3) / e^-x# using the quotient rule?

1 Answer
Nov 2, 2015

# y' = e^x (x^2 + 2x + 3)#
# y'' = e^x (x^2 + 4x + 5)#

Explanation:

Alternatively, you could rewrite the original question as a product and apply the product rule instead of the quotient rule.

STEP 1: Rewrite the original equation
#y = (x^2+3)/e^-x = (x^2+3)*e^x#

STEP 2: Use the product rule to find the first derivative
#y' = (2x)(e^x) + (x^2+3) * e^(-x) #

STEP 3: Simplify your answer to y'
#y' = 2xe^x + x^2e^x + 3e^x = e^x(x^2 + 2x + 3)#

STEP 4: Use the product rule to find the derivative of the derivative, which will give us y''
Recall the product rule is - the derivative of the 1st term, times the 2nd term PLUS the 1st term times the derivative of the 2nd term
In this question, our 1st term is: e^x; our 2nd term is: x^2 + 2x + 3
#y'' = e^x * (x^2 + 2x + 3) + e^x * (2x + 2)#
# = e^x (x^2 + 2x + 3 + 2x + 2)#
# = e^x (x^2 + 4x + 5)#