Welcome to the Toolbox!

In your previous lessons, you learned how to handle simple functions using the Power Rule and the Sum/Difference rules. But what happens when functions start interacting? In the real world, things are rarely isolated. Rates of change often depend on two different things happening at once—like the area of a growing rectangle where both the length and width are changing. This chapter introduces the Product Rule and the Quotient Rule, along with the derivatives for the "other four" trigonometric functions. These are essential tools that you will use in almost every single unit of AP Calculus BC.

Don't worry if these formulas look a bit long at first. With a few catchy rhymes and a bit of practice, they will become second nature!

1. The Product Rule

The Product Rule is used when you need to find the derivative of two functions that are multiplied together.

The Common Trap: Many students want to simply multiply the derivatives together. For example, if \( y = x^2 \cdot \sin(x) \), it is not correct to say the derivative is \( 2x \cdot \cos(x) \). Calculus is a bit more sophisticated than that!

The Formula

If \( h(x) = f(x) \cdot g(x) \), then:

\( \frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x) \)

How to Remember It

Try this mnemonic: "First d-Second plus Second d-First."

  • First: Keep the first function as it is.
  • d-Second: Multiply by the derivative of the second function.
  • Plus: Always add in the product rule.
  • Second: Keep the second function as it is.
  • d-First: Multiply by the derivative of the first function.
Step-by-Step Example

Find the derivative of \( y = x^3 e^x \).

1. Identify the parts: \( f(x) = x^3 \) and \( g(x) = e^x \).
2. Find their individual derivatives: \( f'(x) = 3x^2 \) and \( g'(x) = e^x \).
3. Plug into the formula: \( y' = (x^3)(e^x) + (e^x)(3x^2) \).
4. Simplify if needed: \( y' = x^2 e^x (x + 3) \).

Key Takeaway: When functions multiply, their rates of change combine by taking turns. One stays still while the other differentiates, then they swap!

2. The Quotient Rule

The Quotient Rule is used when one function is divided by another. Because division is not "commutative" (order matters!), this formula is a bit stricter than the Product Rule.

The Formula

If \( h(x) = \frac{f(x)}{g(x)} \), then:

\( \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} \)

How to Remember It

This is the most famous rhyme in Calculus: "Low d-High minus High d-Low, over Low-Low."

  • Low: The denominator \( g(x) \).
  • d-High: The derivative of the numerator \( f'(x) \).
  • Minus: It must be subtraction!
  • High: The numerator \( f(x) \).
  • d-Low: The derivative of the denominator \( g'(x) \).
  • Low-Low: The denominator squared \( (g(x))^2 \).
Common Pitfall Alert!

Because there is a minus sign in the numerator, you must start with the bottom function ("Low"). If you switch them, your answer will have the wrong sign (positive instead of negative or vice versa).

Step-by-Step Example

Find the derivative of \( y = \frac{\ln(x)}{x^2} \).

1. Low: \( x^2 \); High: \( \ln(x) \).
2. d-Low: \( 2x \); d-High: \( \frac{1}{x} \).
3. Apply the rhyme: \( y' = \frac{(x^2)(\frac{1}{x}) - (\ln(x))(2x)}{(x^2)^2} \).
4. Simplify: \( y' = \frac{x - 2x\ln(x)}{x^4} = \frac{1 - 2\ln(x)}{x^3} \).

Quick Review: Product Rule = Plus. Quotient Rule = Minus. Always start with the bottom for Quotient!

3. Derivatives of tan, cot, sec, and csc

In Unit 2.7, you learned that \( \frac{d}{dx}[\sin(x)] = \cos(x) \) and \( \frac{d}{dx}[\cos(x)] = -\sin(x) \). Now it is time to complete the set. You are expected to memorize these for the AP Exam, as no formula sheet is provided.

The Formulas

\( \frac{d}{dx}[\tan(x)] = \sec^2(x) \)

\( \frac{d}{dx}[\cot(x)] = -\csc^2(x) \)

\( \frac{d}{dx}[\sec(x)] = \sec(x)\tan(x) \)

\( \frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x) \)

Memory Tricks to Save Time

1. The "CO" Rule: Notice that the derivative of every "CO" function (Cosine, Cotangent, Cosecant) starts with a negative sign. If it starts with "Co", it’s negative!

2. The Buddy System: Tangent and Secant are "buddies," and Cotangent and Cosecant are "buddies."

  • Derivative of \( \tan \) involves \( \sec \).
  • Derivative of \( \sec \) involves both \( \sec \) and \( \tan \).

Did you know?

You don't actually need to memorize these if you know the Quotient Rule! Since \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), you can use the Quotient Rule to prove that its derivative is \( \sec^2(x) \). However, memorizing them is much faster for the AP Exam's timed sections.

4. Summary Table for Quick Reference

Use this table to quiz yourself:

Rule Name | Function Form | Derivative Appearance
----------------|-------------------|-------------------------
Product Rule | \( f \cdot g \) | \( f g' + g f' \)
Quotient Rule | \( \frac{f}{g} \) | \( \frac{g f' - f g'}{g^2} \)
Tangent | \( \tan(x) \) | \( \sec^2(x) \)
Secant | \( \sec(x) \) | \( \sec(x)\tan(x) \)
Cotangent | \( \cot(x) \) | \( -\csc^2(x) \)
Cosecant | \( \csc(x) \) | \( -\csc(x)\cot(x) \)

Final Pro-Tip for the AP Exam

When solving Free-Response Questions (FRQs), you are often not required to simplify your algebraic answers. If you correctly apply the Product or Quotient rule, you can sometimes leave the expression in its "unsimplified" form to avoid making a basic arithmetic error!

Note: For more complex derivatives involving nested functions, see the next chapter on the Chain Rule.