# GRE Subject Test: Math : Quotient Rule

## Example Questions

### Example Question #8 : Finding Derivatives

Find :

Explanation:

Write the quotient rule.

For the function  and ,  and .

Substitute and solve for the derivative.

Reduce the first term.

### Example Question #9 : Finding Derivatives

Find the following derivative:

Given

Explanation:

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

### Example Question #2 : Finding Derivatives

Find derivative .

Explanation:

This question yields to application of the quotient rule:

So find  and  to start:

### Example Question #1 : Quotient Rule

Find the derivative of: .

None of the Above

Explanation:

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

### Example Question #2 : Quotient Rule

Find the derivative of:

Explanation:

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

### Example Question #3 : Quotient Rule

Find derivative .

Explanation:

This question yields to application of the quotient rule:

So find  and  to start:

### Example Question #4 : Quotient Rule

Find the second derivative of:

None of the Above

None of the Above

Explanation:

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: