The shortest and simplest way to find the derivative of this function is to use the Chain Rule. The Chain Rule definition is  . This is somewhat difficult to read and work with at first. Putting it in words helps though. What this definition states is that the derivative of "layered functions" is the derivative of the outer function times the derivative of the inner function. When I say "layered functions", I mean functions inside other functions. In this problem, we have the function, , inside of a cubic function ,, where is holding the place of the inner function. The outer function is the cubic, while the inner function is the.

Applying the chain rule to this pair of layers means applying the power rule to the outer function, then multiplying it by the derivative of the inner function. Doing so gives

We will need to find the derivative of the inner function, , but first we will write the expression using the actual inner function.

.

To find , we will take the derivative of the two terms inside separately.

The derivative of is . 

The derivative of is another Chain Rule. We take the derivative of outer function, , to get of the same inner function. Then we multiply it by the derivative of the inner function. The derivative of is .

Putting these together we get the following for the derivative of:

Simplifying it, we get

Putting this at the end of the original chain rule we have

This cannot be simplified, so it is the final answer

According to the chain rule   .

Therefore, the derivative we are looking for will be 

Differentiate both sides of the equation: 

Simplify:

Use implicit differentiation to evaluate : 

Simplify:

Subtract  from both sides of the equation: 

Divide both sides of the equation by siny: 

Simplify: 

Solution: 

Differentiate both sides and proceed with the product rule: 

                                                                                      (1)

Evaluate the derivatives in each term. For the first term,  

                                                (2)

 apply the chain rule, 

So now the first term in equation (2) can be written, 

                            (3)

The second term in equation (2) is easy, this is just the product of  multiplied by the derivative of , 

                                                               (4)

Combine equations (3) and (4) to write the derivative, 

Use the product rule to find the derivative. 

Use the power rule to find the derivative.

Thus, the derivative is 

Here we use the product rule: 

Let  and 

Then  (using the chain rule)

and  (using the chain rule)

Subbing these values back into our equation gives us

Simplify by combining like-terms

and pulling out a  from each term gives our final answer

When evaluating the derivative, pay attention to the fact that  are constants, (not variables) and are treated as such.

.

and hence

.

To obtain an expression for , we can take the derivative of  using the sum rule.

.

Substituting  into this equation gives us

.

To find , we will need to use the quotient rule; .

. Start

. Use the quotient rule.

. Take the derivatives inside of the quotient rule. The derivative of  uses the product rule.

. Simplify to match the correct answer.