Inside This Unit: The Full Breakdown
Differentiation measures instantaneous rate of change. This unit defines the derivative as a limit, introduces key derivative rules, and connects derivatives to tangent lines and rates of change.
Why it matters
The derivative is the central concept of differential calculus. Nearly every AP Calculus AB free-response question involves taking a derivative. Mastering the power rule, product rule, quotient rule, and trigonometric derivatives here sets up success for the rest of the course.
Key concepts
- The derivative f'(a) is defined as the limit of [f(a+h) - f(a)] / h as h approaches 0.
- The power rule, product rule, and quotient rule are the fundamental differentiation formulas.
- Derivatives of sin(x), cos(x), e^x, and ln(x) must be memorized.
- Differentiability implies continuity, but continuity does not imply differentiability.
The Derivative as a Limit
The derivative of f at x = a is the limit of the difference quotient as h approaches 0. Geometrically this limit gives the slope of the tangent line at (a, f(a)). The tangent line equation is y - f(a) = f'(a)(x - a). If the limit does not exist — for instance at a corner, cusp, or vertical tangent — the function is not differentiable there. Understanding the limit definition helps you interpret what the derivative actually measures: an instantaneous rate of change.
Basic Derivative Rules
The power rule states that d/dx[x^n] = nx^(n-1) for any real n. The constant multiple rule and sum/difference rule let you differentiate term by term. The product rule d/dx[f*g] = f'g + fg' handles products of functions, while the quotient rule d/dx[f/g] = (f'g - fg') / g^2 handles ratios. These rules combine to differentiate any polynomial, rational, or algebraic expression. Practice applying them quickly and accurately because speed matters on the multiple-choice section.
Derivatives of Transcendental Functions
Several derivative formulas must be committed to memory: d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[tan x] = sec^2 x, d/dx[e^x] = e^x, and d/dx[ln x] = 1/x. These formulas appear in nearly every section of the course going forward. On the AP exam, you will combine these with the product and quotient rules to differentiate expressions like e^x * sin(x) or ln(x)/x^2. Memorize them early so they become automatic.
AP exam tip
When a free-response question says "use the definition of the derivative," you must write the limit of the difference quotient — shortcut rules alone will not earn full credit.
Connections to other units
- Unit 3: The chain rule extends these basic rules to composite functions.
- Unit 4: Rates of change problems apply derivatives to real-world contexts like velocity and related rates.
- Unit 6: Integration reverses differentiation, and every antiderivative rule mirrors a derivative rule.
