Question

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

Answer 1

`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