Question

How do you find the derivative of `cos^7(e^x)` using the chain rule?

Answer 1

`-7e^xcos^6(e^x)sin(e^x)`

Explanation:

The overriding issue in this scenario is the fact that the `cos` function is to the `7th` power.

With the chain rule, we can say that `f'(x)=7cos^6(e^x)*d/(dx)[cos(e^x)]`.

We can use chain rule again to determine `d/(dx)[cos(e^x)]`.
`d/(dx)[cos(e^x)]=-sin(e^x)*d/(dx)[e^x]=-e^xsin(e^x)`

We can plug this back into our equation from earlier:
`f'(x)=7cos^6(e^x)*-e^xsin(e^x)=color(blue)(-7e^xcos^6(e^x)sin(e^x)`