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Calculus/finding the derivative

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Expert: - 10/20/2011


Question
Hi,
I am having a bit of trouble finding the first and second derivative of the function f(x)=xe^x
I think f'(x)=(x)(e^x)+(e^x)(1) which would be rearranged to
e^x(x+1)
I'm completely stuck with the second derivative. Do I just reapply the product rule to get f"(x)=(e^x)(1)+(x+1)(e^x)?
Thanks for your help,
Erin

Answer
The derivative follows the product rule.
If we have f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x).

Yes, that is correct; just reapply the chain rule.

Note that if we take m(x) = (x+k)e^x, then m'(x) = e^x + (x+k)e^x = (x+k+1)e^x.
This means the nth derivative of xe^x is (x+n)e^x.

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