Question

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

Answer 1

`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)`