How do you find the derivative of #f(x) = -5 e^{x \cos x}# using the chain rule?

1 Answer
Jun 4, 2017

Answer:

#d/dxf(x) = -5e^(xcosx)(cosx-xsinx)#

Explanation:

Let's not worry about the #-5# for now. We'll tack that on at the end since it doesn't affect the derivative process at all.

In order to use chain rule, we need to treat #xcosx# like a single variable.

Let #u = xcosx#. Then:

#d/dx(e^(xcosx)) = d/(du)e^u * (du)/dx#

#=e^u*(d/dx(xcosx))#

Now, we need product rule to differentiate:

#=e^(xcosx)*(cosx - xsinx)#

You can simplify this if needed, but this is a fairly good stopping point. All we have to do is multiply by #-5# since the problem originally multiplied by #-5#.

#d/dxf(x) = -5e^(xcosx)(cosx-xsinx)#

Final Answer