How do you differentiate f(x)=1/x*5x*e^x using the product rule?

1 Answer
Aug 28, 2016

5e^x

Explanation:

frac{d}{dx}(frac{1}{x}5xe^x)

taking the constant out, (acdot f)^'=acdot f^'

=5frac{d}{dx}(frac{1}{x}xe^x)

applying product rule,(fcdot g)^'=f^'cdot g+fcdot g^'

=5(frac{d}{dx}(frac{1}{x})xe^x+frac{d}{dx}(xe^x)frac{1}{x})

we know,
frac{d}{dx}(frac{1}{x})=-frac{1}{x^2}
and,
frac{d}{dx}(xe^x)=e^x+e^x*x

finally,
=5((-frac{1}{x^2})xe^x+(e^x+e^x*x)frac{1}{x})

solving it,
5e^x