Question

let f(x)=(x^2)(e^5x)

find the derivative f'(x) and the second derivative, f"(x)

Answer 1

DERIVATE OF 1ST TIMES 2ND PLUS DERIVATIVE OF 2ND TIMES 1ST

Answer 2

PRODUCT RULE

Answer 3

Use the product rule!

f(x) = (x^2)(e^5x)

First times the derivative of the second, plus, second times the derivative of the first:

(d/dx) f(x) = (x^2) d/dx(e^5x) + (e^5x) d/dx (x^2)

what's the derivative of e^5x? well you have to chain rule for that:

d/dx (e^5x) = (e^5x) (5)

now what is the derivative of x^2? yup! just 2

d/dx (x^2) = 2

so what's f'(x)? Let's just plug in our derivatives

f'(x) = (x^2) (e^5x) (5) + (e^5x)(2)

tada!

Answer 4

f'(x) = 2x * e^(5x) + ( 1/5) x² e^(5x)
f"(x) =
[ 2e^(5x) + (2/5)x e^(5x) ]+ [(2/5)x e^(5x) + [(1/25) x² e^(5x) ]

Answer 5

-> f"(x) = e^(5x) [ (x²/ 25) + (4x/5) + 2 ]