Derivatives are the conceptual and computational engine of AP Calculus AB. Every major topic in the second half of the course — related rates, curve sketching, optimization, and applications of integration — depends on fluency with differentiation. The AP exam tests derivatives both directly (compute f′(x)) and embedded inside larger problems, so procedural speed matters as much as conceptual understanding.
This guide covers every core differentiation technique in AP Calculus AB, with worked examples and commentary on the exam-relevant reasoning behind each rule.
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AP Calculus AB Units Involving Derivatives
- Unit 2: Differentiation — Definition and Fundamental Properties
- Unit 3: Differentiation — Composite, Implicit, and Inverse Functions
- Unit 4: Contextual Applications of Differentiation
- Unit 5: Analytical Applications of Differentiation (relative extrema, MVT, L’Hôpital’s Rule)
The Power Rule
The power rule is the most frequently applied differentiation rule. For any function f(x) = xⁿ, the derivative is f′(x) = nxⁿ⁻¹.
Worked Examples
1. f(x) = x⁵
f′(x) = 5x⁴
2. g(x) = 3x⁴ − 7x² + 2x − 9
g′(x) = 12x³ − 14x + 2. The derivative of any constant is 0.
3. h(x) = x⁻³
h′(x) = −3x⁻⁴ = −3/x⁴. Negative exponents follow the same power rule.
4. f(x) = √x = x^(1/2)
f′(x) = (1/2)x^(−1/2) = 1/(2√x). Always rewrite radicals as fractional exponents before differentiating.
5. y = 5/x³ = 5x⁻³
dy/dx = −15x⁻⁴ = −15/x⁴
The Product Rule
For f(x) = u(x) · v(x): f′(x) = u′v + uv′
Worked Examples
6. f(x) = x² · sin x
u = x², u′ = 2x. v = sin x, v′ = cos x. f′(x) = 2x sin x + x² cos x.
7. g(x) = (3x + 1)(x² − 5)
u = 3x + 1, u′ = 3. v = x² − 5, v′ = 2x. g′(x) = 3(x² − 5) + (3x + 1)(2x) = 3x² − 15 + 6x² + 2x = 9x² + 2x − 15.
8. h(x) = eˣ · ln x
u = eˣ, u′ = eˣ. v = ln x, v′ = 1/x. h′(x) = eˣ ln x + eˣ/x = eˣ(ln x + 1/x).
Note: When two expressions are multiplied and neither is a constant, the product rule is required. Students who distribute first and then differentiate avoid the product rule in polynomial cases — but that strategy doesn’t work for transcendental functions like eˣ · ln x.
The Quotient Rule
For f(x) = u(x)/v(x): f′(x) = (u′v − uv′) / v²
Worked Examples
9. f(x) = (x² + 3) / (2x − 1)
u = x² + 3, u′ = 2x. v = 2x − 1, v′ = 2. f′(x) = [2x(2x−1) − (x²+3)(2)] / (2x−1)². Numerator: 4x² − 2x − 2x² − 6 = 2x² − 2x − 6. f′(x) = (2x² − 2x − 6)/(2x−1)².
10. g(x) = sin x / x²
u = sin x, u′ = cos x. v = x², v′ = 2x. g′(x) = (x² cos x − 2x sin x) / x⁴ = (x cos x − 2 sin x) / x³.
Memory tip: “Low d-high minus high d-low, all over low squared.” The order of subtraction is critical — reversing it changes the sign of the answer.
The Chain Rule
For f(x) = g(h(x)): f′(x) = g′(h(x)) · h′(x)
In words: “Derivative of the outside, keep the inside, times derivative of the inside.”
Worked Examples
11. f(x) = (3x + 2)⁵
Outside: u⁵, derivative = 5u⁴. Inside: 3x + 2, derivative = 3. f′(x) = 5(3x+2)⁴ · 3 = 15(3x+2)⁴.
12. g(x) = sin(x³)
Outside: sin(u), derivative = cos(u). Inside: x³, derivative = 3x². g′(x) = cos(x³) · 3x² = 3x² cos(x³).
13. h(x) = e^(2x² − 5)
Outside: eᵘ, derivative = eᵘ. Inside: 2x² − 5, derivative = 4x. h′(x) = e^(2x²−5) · 4x = 4x · e^(2x²−5).
14. f(x) = ln(4x + 1)
Outside: ln(u), derivative = 1/u. Inside: 4x+1, derivative = 4. f′(x) = 1/(4x+1) · 4 = 4/(4x+1).
15. y = √(x² + 9) = (x² + 9)^(1/2)
f′(x) = (1/2)(x² + 9)^(−1/2) · 2x = x/√(x² + 9).
Combined Chain + Product Rule
16. f(x) = x² · e^(3x)
Product rule with chain rule on eˣ. u = x², u′ = 2x. v = e^(3x), v′ = 3e^(3x) (chain rule). f′(x) = 2x·e^(3x) + x²·3e^(3x) = e^(3x)(2x + 3x²) = xe^(3x)(2 + 3x).
Implicit Differentiation
When y is defined implicitly as a function of x (not solved for y), differentiate both sides with respect to x. Every time you differentiate a term involving y, multiply by dy/dx using the chain rule.
Worked Examples
17. Find dy/dx for x² + y² = 25.
Differentiate both sides: 2x + 2y(dy/dx) = 0 → dy/dx = −x/y.
18. Find dy/dx for x³ + 3xy + y² = 7.
Differentiate: 3x² + 3[y + x(dy/dx)] + 2y(dy/dx) = 0. Expand: 3x² + 3y + 3x(dy/dx) + 2y(dy/dx) = 0. Collect dy/dx terms: (3x + 2y)(dy/dx) = −3x² − 3y. dy/dx = −3(x² + y)/(3x + 2y).
19. Find the equation of the tangent line to x² + y² = 100 at the point (6, 8).
dy/dx = −x/y = −6/8 = −3/4. Tangent line: y − 8 = −(3/4)(x − 6) → y = −(3/4)x + 9.5 + 8 = −(3/4)x + 12.5.
Derivatives of Trigonometric Functions
The six standard trig derivatives to memorize:
- d/dx[sin x] = cos x
- d/dx[cos x] = −sin x
- d/dx[tan x] = sec²x
- d/dx[cot x] = −csc²x
- d/dx[sec x] = sec x tan x
- d/dx[csc x] = −csc x cot x
20. f(x) = tan(5x)
Chain rule: f′(x) = sec²(5x) · 5 = 5sec²(5x).
Common AP Calculus AB Derivative Mistakes
- Forgetting the chain rule on composed functions: The derivative of sin(x²) is not just cos(x²). You must multiply by the derivative of the inside: 2x cos(x²).
- Mixing up the quotient rule subtraction order: u′v − uv′ is correct. Writing uv′ − u′v changes every sign in the numerator.
- Differentiating a constant as if it’s a variable: The derivative of a number like 7 or π is 0. This appears in implicit differentiation where constants are part of the equation.
- Not simplifying before differentiating: For rational functions where the numerator can be divided by the denominator, simplifying first can reduce the complexity of the chain or quotient rule application.
- Losing dy/dx in implicit differentiation: Every y-term differentiated with respect to x produces a dy/dx factor via the chain rule. Omitting it is the most common implicit differentiation error.
AP Calculus AB Exam: Derivative Problem Types
On the AP Calculus AB exam, derivatives appear in several question formats:
- Direct computation: “Find f′(x)” — requires applying the correct rule(s)
- Evaluate at a point: “Find the slope of the tangent line at x = 2” — compute f′(x) first, then substitute
- Graph interpretation: “At which values of x does f′(x) = 0?” — reading critical points from a graph of f
- Table problems: Given values of f and g and their derivatives at specific points, compute the derivative of f(g(x)) or f(x)g(x) at a specific point using the product/chain rule
- Related rates: Given a geometric or physical relationship, differentiate implicitly with respect to time t
- Motion problems: Position function s(t) — velocity is s′(t), acceleration is s″(t)
AP Calculus AB Resources
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