In order to find , we need to remember how to find  by using the definition of derivative.

Definition of Derivative:

Now lets apply this to our problem.

Now lets expand the numerator and simplify.

Now factor out an h to get

We can simplify and then evaluate the limit.

The limit definition of a derivative is as follows:

where h represents a very small change in x.

Now, when we use the above formula for the given function, we get

which simplified becomes

The limit definition of a derivative is

where h is a very small change in x.

Using the above formula but with the function given, we get

which simplified becomes

Regardless of the x-value of the function, the derivative will always be 1 (the above contains no x). 

The limit definition of a derivative is

where h represents a very small change in x.

Now, use the above formula for the function given:

which simplified becomes

The limit definition of a derivative is

where h represents a very small change in x.

Now, use the above formula for the function given:

which simplified becomes

Now plug in  into the derivative to get .

The limit definition of the derivative of a function is

where h represents a small change in x.

Now, using the given function, evaluate the limit:

which simplified becomes

.

Now, plug in the given x value into the derivative and we get .

To find dy/dx we must take the derivative of the given function implicitly. Notice the term  will require the use of the Product Rule, because it is a composition of two separate functions multiplied by each other. Every other term in the given function can be derived in a straight-forward manner, but this term tends to mess with many students. Remember to use the Product Rule:

Product Rule: 

Now if we take the derivative of each component of the given problem statement:

Notice that anytime we take the derivative of a term with x involved we place a "dx/dx" next to it, but this is equal to "1".

So this now becomes:

Now if we place all the terms with a "dy/dx" onto one side and factor out we can solved for it:

This is one of the answer choices.

To find dx/dy we must take the derivative of the given function implicitly. Notice the term  will require the use of the Product Rule, because it is a composition of two separate functions multiplied by each other. Every other term in the given function can be derived in a straight-forward manner, but this term tends to mess with many students. Remember to use the Product Rule:

Product Rule: 

Now if we take the derivative of each component of the given problem statement:

Notice that anytime we take the derivative of a term with y involved we place a "dy/dy" next to it, but this is equal to "1".

So this now becomes:

Now if we place all the terms with a "dx/dy" onto one side and factor out we can solved for it:

This is one of the answer choices.

In order to find the first derivative

we must derive both sides of the equation since

From the definition of the derivative of the sine function we have

As such, we have

The limit definition of a derivative is

where h represents a small change in x.

Now, use the above formula for the given function:

which simplified becomes

.