AP Calculus Derivatives: A Complete Guide

Derivatives are the foundation of AP Calculus. Every differentiation technique you master here appears in later topics — related rates, optimization, differential equations, and (in BC) parametric and polar calculus. This guide covers every differentiation rule tested on AP Calculus AB and BC, with exam-relevant context throughout.

1. Basic Differentiation Rules

Power Rule

For f(x) = xⁿ: f'(x) = nxⁿ⁻¹. Applies to all real exponents, including negative and fractional ones. For example: d/dx(x⁻³) = −3x⁻⁴ and d/dx(x^(1/2)) = (1/2)x^(−1/2).

Derivatives of trig functions

• d/dx(sin x) = cos x
• d/dx(cos x) = −sin x
• d/dx(tan x) = sec² x
• d/dx(cot x) = −csc² x
• d/dx(sec x) = sec x tan x
• d/dx(csc x) = −csc x cot x

Exponential and logarithmic derivatives

• d/dx(eˣ) = eˣ
• d/dx(aˣ) = aˣ ln a
• d/dx(ln x) = 1/x
• d/dx(log_a x) = 1/(x ln a)

2. Combination Rules

Product Rule

For h(x) = f(x)g(x): h'(x) = f'(x)g(x) + f(x)g'(x).

Example: d/dx(x² sin x) = 2x sin x + x² cos x.

Quotient Rule

For h(x) = f(x)/g(x): h'(x) = [f'(x)g(x) − f(x)g'(x)] / [g(x)]². A useful memory aid: "low d-high minus high d-low, over low-squared."

Chain Rule

For composite functions h(x) = f(g(x)): h'(x) = f'(g(x)) · g'(x). This is the most commonly applied rule in AP Calculus — any time you have a function inside a function.

Example: d/dx(sin(x³)) = cos(x³) · 3x² = 3x² cos(x³).

A common mistake: forgetting to multiply by the derivative of the inner function (the "× g'(x)" part). On any AP exam, chain rule errors cost more points than any other single differentiation error.

3. Implicit Differentiation

When y is defined implicitly (e.g., x² + y² = 25), differentiate both sides with respect to x. Wherever y appears, treat it as a function of x and apply the chain rule: d/dx(yⁿ) = nyⁿ⁻¹ · (dy/dx).

Example: Differentiate x² + y² = 25. → 2x + 2y(dy/dx) = 0 → dy/dx = −x/y.

Implicit differentiation is required for any curve where y can't be isolated, and it appears on virtually every AP Calculus exam.

4. Related Rates

Related rates problems ask how one changing quantity affects another. The approach: write an equation relating the relevant quantities, then differentiate both sides with respect to time t.

Standard problem types: expanding circles/spheres, sliding ladders, filling tanks. The critical step is writing the correct relationship equation — errors here produce wrong answers even with correct differentiation.

5. Applications of Derivatives

Optimization

Find critical points where f'(x) = 0 or f'(x) is undefined. Use the First Derivative Test (sign change of f') or Second Derivative Test (sign of f'') to classify as max or min. For closed interval problems, also check endpoints.

Mean Value Theorem (MVT)

If f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) where f'(c) = [f(b)−f(a)]/(b−a). AP FRQ justification questions frequently require invoking the MVT by name with its conditions stated.

Concavity and inflection points

f''(x) > 0 → concave up; f''(x) < 0 → concave down. An inflection point is where concavity changes — f'' changes sign. Note: f''(c) = 0 alone does not guarantee an inflection point.

For expert support building derivative fluency, our AP Calculus AB tutoring online works through all differentiation techniques with AP-style problems and FRQ practice. BC students preparing for parametric and polar derivatives can get the same level of support through our AP Calculus BC programme.