How do you differentiate #f(x)=1/x5xe^x# using the product rule?

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Answer 1 · 2016-08-28T10:31:47

#5e^x#

#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#

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