AP Calculus AB · Lesson review · September 18, 2026

Product and Quotient Rules

Prove the rules from the derivative definition, apply them to polynomial and trigonometric products, and derive the tangent and secant rules.

38-minute edited lesson10 chaptersCaptions + corrected transcript

01 · Watch

Lesson video

VTT captions SRT captions

0:0001Product and Quotient Rules 2:5802Why Products Need a New Rule 3:4803State the Product Rule 6:0304Prove the Product Rule 14:5305Apply the Product Rule 17:1506Products with Three Factors 25:0207The Quotient Rule 27:0208Apply the Quotient Rule 30:1209Derive the Tangent Rule 34:0210Derive the Secant Rule

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02 · Understand

Learning targets

Recognize when a function is a product or quotient of differentiable functions.
Use and explain the product rule $(fg)'=f'g+fg'$ .
Use and explain the quotient rule, including the denominator squared.
Differentiate products with more than two factors by applying the product rule repeatedly.
Derive the tangent and secant derivative formulas from quotient identities.

03 · Rules

The derivative rules

These rules combine functions without pretending the derivative distributes over multiplication or division.

01Product rule: $\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$ Differentiate the first factor, then the second factor, and add the two products.
02Quotient rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}$ Use low d-high minus high d-low, over low squared.
03Tangent: $\frac{d}{dx}(\tan x)=\sec^2x$ Write tangent as sine over cosine, then simplify with the Pythagorean identity.
04Secant: $\frac{d}{dx}(\sec x)=\sec x\tan x$ Write secant as one over cosine and apply the quotient rule.

04 · Derive and apply

Worked examples

Name the factors first, apply the correct rule, and simplify afterward.

Why it works

Product rule from the definition

h'(x)=\lim_{h\to0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}

Add and subtract the same middle expression so the numerator separates into two recognizable difference quotients.

h'(x)=f'(x)g(x)+f(x)g'(x)

Product rule

Polynomial times sine

f(x)=x^4\sin x

Keep each original factor once while differentiating the other.

f'(x)=4x^3\sin x+x^4\cos x

Three factors

Apply the product rule repeatedly

f(x)=x^5\cos x\,e^x

Differentiate one factor at a time. The final derivative has three terms.

f'(x)=5x^4\cos x\,e^x-x^5\sin x\,e^x+x^5\cos x\,e^x

Quotient rule

Polynomial over sine

f(x)=\frac{x^3}{\sin x}

Differentiate the numerator and denominator, preserve the subtraction order, and square the full denominator.

f'(x)=\frac{3x^2\sin x-x^3\cos x}{\sin^2x}

Trig proof

Derivative of tangent

\frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)=\frac{\cos^2x+\sin^2x}{\cos^2x}

The numerator becomes $\cos^2x+\sin^2x=1$ .

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

Trig proof

Derivative of secant

\frac{d}{dx}\left(\frac1{\cos x}\right)=\frac{\sin x}{\cos^2x}=\frac1{\cos x}\frac{\sin x}{\cos x}

Rewrite the result as a product of secant and tangent.

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

05 · Avoid

Common mistakes

Multiplying the derivatives.

$(fg)'$ is not $f'g'$ . The product rule has two added terms.

Dropping an unchanged factor.

Each product-rule term contains one derivative and one original factor.

Reversing the quotient numerator.

Keep the order $f'g-fg'$ .

Forgetting the square.

The denominator of the quotient-rule result is $[g(x)]^2$ .

Stopping before simplifying.

Use identities such as $\sin^2x+\cos^2x=1$ to reveal standard trig derivatives.

Using only two terms for three factors.

A three-factor product produces three derivative terms.

06 · Check yourself

Quick check

Try each before revealing the answer.

1Differentiate

x^4\sin x

$4x^3\sin x+x^4\cos x$ .

2Which factor is squared in the quotient-rule denominator?

The original denominator function: $[g(x)]^2$ .

3Differentiate

\tan x

$\sec^2x$ .

4Differentiate

\sec x

$\sec x\tan x$ .

5Why is

(fg)'=f'g'

incorrect?

Changing either factor changes the product. The correct derivative adds both contributions: $f'g+fg'$ .

07 · Revisit

Transcript and practice

Search the corrected transcript for a rule or example. Captions follow the edited video timeline.

Download corrected transcript

Return to the Friday 9/18 eKadence activity for the assigned practice.