Computation of the derivative requires the use of the Product Rule and Chain Rule. 

The Product Rule is used in a scenario when one has two differentiable functions multiplied by each other:

This can be easily stated in words as: "First times the derivative of the second, plus the second times the derivative of the first."

In the problem statement, we are given:

 is the "First" function, and  is the "Second" function. 

The "Second" function requires use of the Chain Rule. 

When:

Applying these formulas results in:

Simplifying the terms inside the brackets results in:

We notice that there is a common term that can be factored out in the sets of equations on either side of the "+" sign. Let's factor these out, and make the equation look "cleaner".

Inside the brackets, it is possible to clean up the terms into one expanded function. Let us do this:

Simplifying this results in one of the answer choices:

Recall that one definition for the derivative of a function  is .

This means that this question is asking us to find the value of the derivative of  at .

Since 

 and , the value of the limit is .

Evaluation of this integral requires use of the Product Rule. One must also need to recall the form of the derivative of .

Product Rule:

Applying these two rules results in:

This matches one of the answer choices.

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.

We can simplify this to

Now factor out an h to get

We can simplify and then evaluate the limit.

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.

We can simplify this to

Now factor out an h to get

We can simplify and then evaluate the limit.

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 .