How do you differentiate #f(x)=(1/(3x))xe^x-x*cosx# using the product rule?

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Answer 1 · 2016-12-13T15:31:34

Please see below.

Simplify first

#f(x) = 1/3e^x-xcosx#

We do not need the product rule for the first term, although you may use it if you like.

#d/dx(1/3e^x) = d/dx(1/3)e^x+1/3 d/dx(e^x) = 0e^x + 1/3e^x = 1/3e^x#

For the second term, we need the product rule. (unless we do something unusual.)

#d/dx(xcosx) = d/dx(x) * cosx + x * d/dx(cosx)#

# = cosx-xsinx#

Putting it all together,

#f'(x) = 1/3e^x + cosx - xsinx#

How do you differentiate #f(x)=(1/(3x))*x*e^x-x*cosx# using the product rule?