Derivatives & Integrals - GRE Quantitative Reasoning

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

This problem requires the product rule. The derivative using the product rule is as follows:

Let and .

Their derivatives are:

and

Substitute the functions into the product rule formula and simplify.

Recall the chain rule from calculus:

So we will want to begin by finding the first derivative of each of our functions:

Next, use the chain rule formula:

Expand everything to get

Recall the product rule for differentiaion.

So we need to find the first derivative of each of our functions:

Recall that is a strange one:

Next, use the formula from up above.

Expand and simplify.

Rewrite in standard form and factor out an .

Step 1: We need to define the product rule. The product rule says . is defined as the derivative of f(x) multiplied by g(x). In the question, the first parentheses is f(x) and the second parentheses is g(x).

Step 2: We will first calculate and . To find the derivative of any term, we do one the following rules:

1. All terms with exponents (positive and negative) of the original equation are dropped down and multiplied by the coefficient of that term that you are working on. The exponent that gets written after taking the derivative is 1 less than (can also be thought of as (x-1, where x is the exponent that was dropped).
2. The derivative of a term ax, , is just the value , which is the coefficient of the term.
3. The derivative of any constant term, that is any term that does not have a variable next to it, is always .

Step 3: We will take derivative of first.

. Let us take the derivative of each and every term and then add everything back together. We denote (') as derivative.

. We are using rule 1 here (listed above). The two from the exponent dropped down and was multiplied by the coefficient of that term. The exponent is 1 less than the exponent that was dropped down, which is why you see in the exponent.
. We use rule 2 that was listed above. The derivative of this term is just the coefficient of that term, in this case, 3.
. We use rule 3. Since the derivative is , we won't write it in the final equation for the derivative of .

Let's put everything together:

Step 4: We will take derivative of g(x).

So, .

Step 5: Now that we have found the derivatives, let's substitute all the equations into the formula for product rule.

Step 6: Let's find .

In the equation above, . We will need to distribute and expand this multiplication.

When we expand, we get . Let's simplify that expansion.

We will get: .

Step 7: Let's find , which is defined as .

We will expand and simplify.

When we expand, we get: .

If we simplify, we get .

Step 8: Add the two products together and simplify.

If we add and simplify, we get:

. This is the final answer to the expansion of the product rule.

Step 1: Define and :

Step 2: Find and .

Step 3: Define Product Rule:

Product Rule=.

Step 4: Substitute the functions for their places in the product rule formula

Step 5: Expand:

Step 6: Combine like terms:

Final answer:

Step 1: Define the two functions...

Step 2: Find the derivative of each function:

Step 3: Define the Product Rule Formula...

Step 4: Plug in the functions:

Step 5: Expand and Simplify:

The derivative of the product of and is .

1. Starting Equation:

2. Simplifying:

3. Take the derivative, using the power rule.

1. Simplify answer:

Notes:

1. Easiest way to take the derivative is to simplify the equation first. In doing so, you should see that this is NOT an application for the chain rule. Although two variables are multiplied together, they are the same variable. The chain rule will give you the wrong answer.

2. Exponent Math... multiplied factors means you should add the exponents

3. Standard Power Rule. Bring the exponent down, multiplying it into the coefficient. Subtract 1 from the exponent. Constants go to 0.

4. Simplify the answer.

Write the quotient rule.

For the function , and , and .

Substitute and solve for the derivative.

Reduce the first term.

This question asks us to find the derivative of a quotient. Use the quotient rule:

Start by finding and .

So we get:

Whew, let's simplify

This question yields to application of the quotient rule:

So find and to start:

So our answer is:

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

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 yields to application of the quotient rule:

So find and to start:

So our answer is:

Integration by parts follows the formula:

In this problem we have so we'll assign our substitutions:

and

which means and

Including our substitutions into the formula gives us:

We can pull out the fraction from the integral in the second part:

Completing the integration gives us:

Integration by parts follows the formula:

So, our substitutions will be and

which means and

Plugging our substitutions into the formula gives us:

Since , we have:

, or