How do you find the slope of the tangent line to the graph of the function #f(x)= x*e^x# at x=7?

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Answer 1 · 2018-04-08T14:20:33

#8e^7#

We are given: #f(x)=xe^x#

To find the slope of the tangent line, we plug in #f'(x)# for #x=7#, i.e. #f'(7)#.

So, we differentiate using the product rule, which states that,

#d/dx(f(x)g(x))=f'(x)g(x)+f(x)g'(x)#

And so, we got:

#f'(x)=(x)'e^x+x(e^x)'#

#=1*e^x+xe^x#

#=xe^x+e^x#

#=e^x(x+1)#

We plug in #x=7#, and get,

#f'(7)=e^7(7+1)#

#=8e^7#