How do you differentiate #f(x)=csce^(4x)# using the chain rule.?

Answer 1

#f'(x)=-4e^(4x)csc(e^(4x))cot(e^(4x))#

The chain rule states that when differentiating a function inside of a function, (1) differentiate the outside function and leave the inside function as is, and (2) multiply this by the derivative of the inside version.

In #f(x)=csc(e^(4x))#, we have the outside function #csc(u)# and the inside function #e^(4x)#.

Thus, since the derivative of #csc(x)# is #-csc(x)cot(x)#, we will have

#f'(x)=-csc(e^(4x))cot(e^(4x))*d/dx(e^(4x))#

Don't forget to multiply by the derivative of the inside function, which is #d/dx(e^(4x))#.

To differentiate #e^(4x)#, we will have to use the chain rule again.

The outside function is #e^u# and the inside function is #u=4x#. Since the derivative of #e^x# is still #e^x#, we have

#d/dx(e^(4x))=e^(4x)*d/dx(4x)#

Note that #d/dx(4x)=4#, so we know that

#d/dx(e^(4x))=4e^(4x)#

Plug this back in to the derivative of the whole function:

#f'(x)=-4e^(4x)csc(e^(4x))cot(e^(4x))#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.