Answered

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


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

Verified Answer :

biancamilburn3

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!

HomertheGenius

f ( x ) = ( x² ) · e^(5x) + x² · ( e^(5x) ) =
= 2 x · e^(5x) + 5 e^(5x) · x² =
= x e^(5x) ( 2 + 5 x )
f ( x ) = ( 2 x e^(5x) + 5 x² e^(5x) ) =
= ( 2 x ) ˙e^(5x) + 2 x ( e^(5x) ) + ( 5 x² ) · e^(5x) + ( e^(5x)) ` · 5 x² = 
= 2 · e^(5x) + 10 x · e^(5x) + 10 x · e^(5x) + 25 x² · e^(5x) =
= e^(5x) · ( 2 + 20 x + 25 x² ) 

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