Product and quotient rules; derivatives of tan, cot, sec and csc

Welcome to the Derivative Toolkit!

So far, you’ve learned how to find the derivative of simple functions using the Power Rule and basic trig functions like \(\sin x\) and \(\cos x\). But what happens when functions start "hanging out" together? In this chapter, we explore how to differentiate functions that are multiplied together (The Product Rule) or divided by one another (The Quotient Rule). We will also complete your "trigonometric library" by learning the derivatives for \(\tan x\), \(\cot x\), \(\sec x\), and \(\csc x\).

Think of these rules as the "Swiss Army Knife" of calculus—once you master them, you can handle almost any function the AP exam throws at you!

2.8 The Product Rule

When you see two functions multiplied together, such as \(f(x) = x^2 \sin x\), you cannot simply multiply their individual derivatives. Instead, we use the Product Rule.

The Formula

If \(h(x) = f(x) \cdot g(x)\), then the derivative is:
\(h'(x) = f'(x)g(x) + f(x)g'(x)\)

A Friendly Mnemonic

Many students find it easiest to remember this as: "Left d-Right plus Right d-Left."
In this phrase, "Left" is your first function and "Right" is your second function. The "d" stands for derivative.

Step-by-Step Example

Find the derivative of \(y = x^3 e^x\).
1. Identify the parts: Let \(f(x) = x^3\) and \(g(x) = e^x\).
2. Find their individual derivatives: \(f'(x) = 3x^2\) and \(g'(x) = e^x\).
3. Assemble using the rule: \(y' = (3x^2)(e^x) + (x^3)(e^x)\).
4. Simplify if needed: \(y' = e^x(3x^2 + x^3)\).

Key Takeaway: Always keep the two terms separate and add them together. Order doesn't strictly matter for the Product Rule because of addition, but staying consistent helps prevent errors!

2.9 The Quotient Rule

The Quotient Rule is used when you have one function divided by another, like \(y = \frac{x^2 + 1}{\cos x}\). This rule is a bit more complex, so pay close attention to the order!

The Formula

If \(h(x) = \frac{f(x)}{g(x)}\), then:
\(h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}\)

The "Rhyme" to Remember

This is the most famous mnemonic in calculus. Call the top function "High" and the bottom function "Low":
"Low d-High minus High d-Low, square the bottom and away we go!"

• Low: \(g(x)\)
• d-High: \(f'(x)\)
• minus: \(-\)
• High: \(f(x)\)
• d-Low: \(g'(x)\)
• Square the bottom: \([g(x)]^2\)

Common Mistakes to Avoid

1. Mixing up the order: Because there is a minus sign in the numerator, you must start with the bottom function times the derivative of the top.
2. Forgetting the denominator: Students often get so focused on the numerator that they forget to write the \([g(x)]^2\) at the bottom.
3. Sign errors: Be very careful with parentheses after the minus sign!

Key Takeaway: Order matters! Always start with the bottom function: Low d-High...

2.10 Derivatives of Tan, Cot, Sec, and Csc

In Topic 2.7, you learned that the derivative of \(\sin x\) is \(\cos x\) and the derivative of \(\cos x\) is \(-\sin x\). Now, we add the final four trigonometric derivatives to your memory bank. These can actually be derived using the Quotient Rule, but for the AP exam, you should memorize them to save time.

The "Fab Four" Formulas

\( \frac{d}{dx} \tan x = \sec^2 x \)
\( \frac{d}{dx} \sec x = \sec x \tan x \)
\( \frac{d}{dx} \cot x = -\csc^2 x \)
\( \frac{d}{dx} \csc x = -\csc x \cot x \)

Patterns to Help You Remember

Don't worry if these look intimidating! Look for these three patterns:
1. The "Co-" Rule: Every trig function starting with "co" (\(\cos\), \(\cot\), \(\csc\)) has a derivative that is negative.
2. Tangent and Secant are Besties: The derivative of \(\tan x\) involves \(\sec x\), and the derivative of \(\sec x\) involves both \(\sec x\) and \(\tan x\).
3. Cotangent and Cosecant are Besties: They follow the exact same pattern as tangent/secant, just with "co" and a negative sign.

Did you know? You can prove \(\frac{d}{dx} \tan x = \sec^2 x\) by writing \(\tan x\) as \(\frac{\sin x}{\cos x}\) and applying the Quotient Rule. Try it for practice!

Key Takeaway: If it starts with "C", the derivative is negative. Tangent goes with Secant; Cotangent goes with Cosecant.

Quick Review Checklist

Before moving on to the Chain Rule (Unit 3.1), make sure you can confidently do the following:

• Identify when a function is a product vs. a quotient.
• Recite the "Low d-High" rhyme without looking.
• Recall the derivative of \(\sec x\) and \(\tan x\) instantly.
• Remember that \(\frac{d}{dx} \cot x\) and \(\frac{d}{dx} \csc x\) are negative.

Pro-Tip for the AP Exam: On Free-Response Questions (FRQs), you often do not need to simplify your algebraic answer. If you apply the Quotient Rule correctly, you can leave the giant fraction as it is unless the question asks for a specific value or further proof!