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