How do you differentiate $f (x) = x e^{x} - x$ $f(x)=xe^x-x$ using the product rule?

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Answer 1 · 2015-11-20T23:59:23

$f'(x)=e^x+xe^x-1$

Let's first examine just the $xe^x$ portion of this function.

Through the product rule,
$d/dx[xe^x]=color(blue)(d/dx[x])*e^x+color(purple)(d/dx[e^x]*)x$

Therefore,
$d/dx[xe^x]=color(blue)(1)*e^x+color(purple)(e^x)*x=e^x+xe^x$

We can use this in the original equation to determine that
$f'(x)=e^x+xe^x-d/dx[x]=color(green)(e^x+xe^x-1$

How do you differentiate f(x)=xe^x-xf(x)=xex−xf(x)=xe^x-x using the product rule?