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

1 Answer
Nov 16, 2015

Answer:

#-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)#