# What is the derivative of xe^(2x)?

• February 23rd 2013, 04:17 AM
remutyte
What is the derivative of xe^(2x)?
What is the derivative of x*e^(2x)?

I know that I have to use integration by parts. However, I am not sure if I got the right answer.

Thank you!
• February 23rd 2013, 05:07 AM
MINOANMAN
Re: What is the derivative of xe^(2x)?
The first derivative is e^(2x)[1+2x] and the second is 4e^(2x)[x+1]
• February 26th 2013, 01:06 AM
johnykeets
Re: What is the derivative of xe^(2x)?
As far as my knowledge is concerned, The derivative can be calculated by

using d/dx(uv) = u dv/dx + v du/dx

d/dx(xe^2x) = x d/dx(e^2x) + e^2x as d/dx x = 1

= 2x e^2x + e^2x as d/dx(e^y) = e^y dy/dx [using chain rule]
Therefore, the derivative of xe^(2x)=1.
Hope this is useful.

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• February 26th 2013, 03:16 AM
HallsofIvy
Re: What is the derivative of xe^(2x)?
Quote:

Originally Posted by remutyte
What is the derivative of x*e^(2x)?

I know that I have to use integration by parts. However, I am not sure if I got the right answer.

Thank you!

What, exactly is the question? You first ask about the derivative but then refer to integration.

If the question is about the derivative, you use the product rule: $(xe^x)'= (x)'e^x+ x(e^x)'= e^x+ xe^x= (x+ 1)e^x$.

If the question is about the integral, you use integration by parts. Let u= x, $dv= e^xdx$. Then du= dx, $v= e^x$.
$\int xe^x dx= xe^x- \int e^x dx= xe^x- e^x+ C= (x- 1)e^x+ C$.