### All AP Calculus AB Resources

## Example Questions

### Example Question #1 : Understanding The Limiting Process.

Find the derivative.

y = sec (5x^{3})

**Possible Answers:**

y' = –csc(5x^{3})cot(5x^{3})(15x^{2})

y' = –sec(5x^{3})tan(5x^{3})(15x^{2})

y' = sec(5x^{3})tan(5x^{3})

y' = –csc(5x^{3})cot(5x^{3})

y' = sec(5x^{3})tan(5x^{3})(15x^{2})

**Correct answer:**

y' = sec(5x^{3})tan(5x^{3})(15x^{2})

The derivative of the function y = sec(x) is sec(x)tan(x). First take the derivative of the outside of the function: y = sec(4x^{3}) : y' = sec(5x^{3})tan(5x^{3}). Then take the derivative of the inside of the function: 5x^{3} becomes 15x^{2}. So your final answer is: y' = ec(5x^{3})tan(5x^{3})15x^{2}

### Example Question #1 : Understanding The Limiting Process.

Find the slope of the tangent line to the graph of *f* at *x* = 9, given that f(x) = –x^{2 }+ 5√(x)

**Possible Answers:**

–18 – (5/6)

–18 + (5/6)

–18

18

18 + (5/6)

**Correct answer:**

–18 + (5/6)

First find the derivative of the function.

f(x) = –x^{2} + 5√(x)

f'(x) = –2x + 5(1/2)x^{–1/2}

Simplify the problem

f'(x) = –2x + (5/2x^{1/2})

Plug in 9.

f'(3) = –2(9) + (5/2(9)^{1/2})

= –18 + 5/(6)

### Example Question #2 : Understanding The Limiting Process.

Find the derivative

(x + 1)/(x – 1)

**Possible Answers:**

1

(x + 1) + (x – 1)

(–2)/(x – 1)

(–2)/(x + 1)^{2}

(–2)/(x – 1)^{2}

**Correct answer:**

(–2)/(x – 1)^{2}

Rewrite problem.

(x + 1)/(x – 1)

Use quotient rule to solve this derivative.

((x – 1)(1) – (x + 1)(1))/(x – 1)^{2}^{}

(x – 1) – x – 1)/(x – 1)^{2}

–2/(x – 1)^{2}

### Example Question #2 : Understanding The Limiting Process.

**Possible Answers:**

**Correct answer:**

Use the chain rule and the formula

### Example Question #5 : Understanding The Limiting Process.

Find the derivative of

**Possible Answers:**

**Correct answer:**

The answer is . It is easy to solve if we multiply everything together first before taking the derivative.

### Example Question #3 : Understanding The Limiting Process.

Differentiate .

**Possible Answers:**

**Correct answer:**

Using the power rule, multiply the coefficient by the power and subtract the power by 1.

### Example Question #4 : Understanding The Limiting Process.

Differentiate .

**Possible Answers:**

**Correct answer:**

Use the product rule:

### Example Question #5 : Understanding The Limiting Process.

Differentiate:

**Possible Answers:**

**Correct answer:**

Use the product rule to find the derivative of the function.

### Example Question #6 : Understanding The Limiting Process.

Differentiate:

**Possible Answers:**

**Correct answer:**

The derivative of any function of e to any exponent is equal to the function multiplied by the derivative of the exponent.

### Example Question #8 : Understanding The Limiting Process.

Find the second derivative of .

**Possible Answers:**

**Correct answer:**

Factoring out an x gives you .

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