AP Calculus Formula Sheet

Differentiation Rules

Basic Rules

• Power Rule: d/dx(xⁿ) = nxⁿ⁻¹
• Constant: d/dx(c) = 0
• Sum/Difference: (f ± g)' = f' ± g'
• Constant Multiple: (cf)' = cf'
• Product Rule: (fg)' = f'g + fg'
• Quotient Rule: (f/g)' = (f'g − fg') / g²
• Chain Rule: (f∘g)' = f'(g(x)) · g'(x)

Derivatives of Special 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
• 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)
• d/dx(arcsin x) = 1/√(1−x²)
• d/dx(arccos x) = −1/√(1−x²)
• d/dx(arctan x) = 1/(1+x²)

Integration Rules

Basic Antiderivatives

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
• ∫1/x dx = ln|x| + C
• ∫eˣ dx = eˣ + C
• ∫aˣ dx = aˣ/ln a + C
• ∫sin x dx = −cos x + C
• ∫cos x dx = sin x + C
• ∫sec² x dx = tan x + C
• ∫csc² x dx = −cot x + C
• ∫sec x tan x dx = sec x + C
• ∫csc x cot x dx = −csc x + C
• ∫1/√(1−x²) dx = arcsin x + C
• ∫1/(1+x²) dx = arctan x + C

Integration Techniques (AB and BC)

• u-substitution: ∫f(g(x))g'(x) dx — let u = g(x)
• Integration by parts (BC): ∫u dv = uv − ∫v du
• Partial fractions (BC): decompose rational functions before integrating
• Improper integrals (BC): ∫_a^∞ f(x) dx = lim_{b→∞} ∫_a^b f(x) dx

Key Theorems

• Fundamental Theorem of Calculus (Part 1): If F'(x) = f(x), then ∫_a^b f(x) dx = F(b) − F(a)
• FTC Part 2: d/dx [∫_a^x f(t) dt] = f(x)
• Mean Value Theorem: If f is continuous on [a,b] and differentiable on (a,b), ∃ c ∈ (a,b) where f'(c) = [f(b)−f(a)]/(b−a)
• Intermediate Value Theorem: If f is continuous on [a,b] and k is between f(a) and f(b), ∃ c ∈ (a,b) where f(c) = k
• Extreme Value Theorem: A continuous function on a closed interval attains its max and min
• Rolle's Theorem: If f(a) = f(b) and f is differentiable on (a,b), ∃ c where f'(c) = 0

Applications of Integration

• Area between curves: A = ∫_a^b [f(x) − g(x)] dx, where f(x) ≥ g(x)
• Volume — disk method: V = π ∫_a^b [f(x)]² dx (rotation about x-axis)
• Volume — washer method: V = π ∫_a^b ([f(x)]² − [g(x)]²) dx
• Volume — shell method (BC): V = 2π ∫_a^b x·f(x) dx (rotation about y-axis)
• Arc length (BC): L = ∫_a^b √(1 + [f'(x)]²) dx
• Average value of a function: f_avg = (1/(b−a)) ∫_a^b f(x) dx

Series and Convergence (BC only)

Common Series Tests

• Geometric series: Σ arⁿ converges to a/(1−r) if |r| < 1
• p-series: Σ 1/nᵖ converges if p > 1
• Integral Test: Σ f(n) and ∫f(x)dx have same convergence behaviour
• Comparison Test: If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges
• Limit Comparison: If lim aₙ/bₙ = L (finite, positive), same convergence
• Ratio Test: lim |aₙ₊₁/aₙ| < 1 → converges; > 1 → diverges
• Alternating Series Test: Σ (−1)ⁿbₙ converges if bₙ decreasing → 0

Taylor and Maclaurin Series

• eˣ = Σ xⁿ/n! = 1 + x + x²/2! + x³/3! + …
• sin x = Σ (−1)ⁿ x^(2n+1)/(2n+1)! = x − x³/6 + x⁵/120 − …
• cos x = Σ (−1)ⁿ x^(2n)/(2n)! = 1 − x²/2 + x⁴/24 − …
• 1/(1−x) = Σ xⁿ = 1 + x + x² + … (|x| < 1)
• ln(1+x) = Σ (−1)ⁿ⁺¹ xⁿ/n = x − x²/2 + x³/3 − … (|x| ≤ 1, x ≠ −1)