### All GRE Subject Test: Math Resources

## Example Questions

### Example Question #1 : Quotient Rule

Find :

**Possible Answers:**

**Correct answer:**

Write the quotient rule.

For the function , and , and .

Substitute and solve for the derivative.

Reduce the first term.

### Example Question #2 : Quotient Rule

Find the following derivative:

Given

**Possible Answers:**

**Correct answer:**

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 #3 : Quotient Rule

Find derivative .

**Possible Answers:**

**Correct answer:**

This question yields to application of the quotient rule:

So find and to start:

So our answer is:

### Example Question #4 : Quotient Rule

Find the derivative of: .

**Possible Answers:**

None of the Above

**Correct answer:**

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 #5 : Quotient Rule

Find the derivative of:

**Possible Answers:**

**Correct answer:**

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 #6 : Quotient Rule

Find derivative .

**Possible Answers:**

**Correct answer:**

This question yields to application of the quotient rule:

So find and to start:

So our answer is:

### Example Question #7 : Quotient Rule

Find the **second** derivative of:

**Possible Answers:**

None of the Above

**Correct answer:**

None of the Above

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...

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