Step 1: We need to define the quotient rule. The quotient rule says:, whereis the derivative ofandis 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 islower 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: Findand:




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 ofis

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 findandto 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 anout from the numerator and denominator. Simplify the fraction..

We have found the first derivative..

Finding Second Derivative:

Step 6: Findfrom 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 outand 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 areand.

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 ininto 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 ininto the reduced factor form:

Simplify as much as possible...

The limit of this function as x approachesis