Step 1: We need to define the quotient rule. The quotient rule says: , where  is the derivative of  and  is the derivative of 

Step 2: We need to review how to take derivatives of different kinds of terms. When we are taking the derivative of terms in a polynomial, we need to follow these rules:

Rule 1: For any term with an exponent, the derivative of that term says: Drop the exponent and multiply it to the coefficient of that term. The new exponent of the derivative is  lower than the previous exponent. 

Example: 

Rule 2: For any term in the form , the derivative of that term is just , the coefficient of that term.

Ecample: 

Rule 3: The derivative of any constant is always 

Step 3: Find  and :




Step 4: Plug in all equations into the quotient rule:



Step 5: Simplify the fraction in step 4:



Step 6: Combine terms in the numerator in step 5:

.

The derivative of  is 

Step 1: Define .



Step 2: Find .



Step 3: Plug in the functions/values into the formula for quotient rule: 



The derivative of the expression is 

This question yields to application of the quotient rule:

So find  and  to start:

So our answer is:

Finding the First Derivative:

Step 1: Define 



Step 2: Find 



Step 3: Plug in all equations into the quotient rule formula: 



Step 4: Simplify the fraction in step 3:






Step 5: Factor an  out from the numerator and denominator. Simplify the fraction..

We have found the first derivative..

Finding Second Derivative:

Step 6: Find  from the first derivative function



Step 7: Find 



Step 8: Plug in the expressions into the quotient rule formula: 

Step 9: Simplify:

I put "..." because the numerator is very long. I don't want to write all the terms...

Step 10: Combine like terms:



Step 11: Factor out  and simplify:

Final Answer: .

This is the second derivative.


The answer is None of the Above. The second derivative is not in the answers...

This question requires application of multiple chain rules.  There are 2 inner functions in , which are  and .  

The brackets are to identify the functions within the function where the chain rule must be applied.

Solve the derivative.

The sine of sine of an angle cannot be combined to be sine squared.

Therefore, the answer is: 

Recall chain rule for this problem

So if we are given the following,

We can think of it like this

Clean it up a bit to get:

Chain Rule:

For this problem

Plug the values into the Chain Rule formula and simplify:

Step 1: Try plugging in  into the denominator of the function. We want to make sure that the bottom does not become ...

.. We got zero, and we cannot have zero in the denominator. So, we must try and factor the function (numerator and denominator):

Step 2: Factor:



Step 3: Reduce:



Step 4: Now that we got rid of the factor that made the denominator zero, we know that this function has a limit.

Step 5: Plug in  into the reduced factor form:

Simplify as much as possible...

The limit of this function as x approaches  is 

The first step is to differentiate both sides with respect to :

Note: Those that are functions of  can be differentiated with respect to , just remember to mulitply it by 

Now we can solve for :

Our first step would be to differentiate both sides with respect to :

The functions of  can be differentiated with respect to , just remember to multiply by  .