AP Calculus AB Formula Sheet

Limit Laws

lim [f(x) ± g(x)] = lim f(x) ± lim g(x), lim [f(x) · g(x)] = lim f(x) · lim g(x), lim [f(x)/g(x)] = lim f(x) / lim g(x), if lim g(x) ≠ 0

Continuity at x = a

lim x→a f(x) = f(a)

Point-slope form

y - f(a) = f'(a)(x - a)

Power Rule

d/dx [x^n] = n·x^(n-1)

Constant Rule

d/dx [c] = 0

Constant Multiple Rule

d/dx [c·f(x)] = c·f'(x)

Sum/Difference Rule

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Product Rule

(fg)' = f'·g + f·g'

Quotient Rule

(f/g)' = (f'·g - f·g') / g²

Chain Rule

d/dx [f(g(x))] = f'(g(x))·g'(x)

Trig Derivatives: sin x

d/dx [sin x] = cos x

Trig Derivatives: cos x

d/dx [cos x] = -sin x

Trig Derivatives: tan x

d/dx [tan x] = sec² x

Trig Derivatives: sec x

d/dx [sec x] = sec x·tan x

Trig Derivatives: csc x

d/dx [csc x] = -csc x·cot x

Trig Derivatives: cot x

d/dx [cot x] = -csc² x

Implicit Differentiation

d/dx [y²] = 2y·dy/dx

Indefinite Integrals

∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C, ∫ e^x dx = e^x + C, ∫ a^x dx = a^x / ln(a) + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C

FTC Part 2

∫ from a to b of f(x) dx = F(b) - F(a)

FTC Part 1

d/dx [∫ from a to x of f(t) dt] = f(x)

Critical Points

f'(x) = 0 or undefined

Increasing/Decreasing

Increasing: f'(x) > 0, Decreasing: f'(x) < 0

Concavity

Concave Up: f''(x) > 0, Concave Down: f''(x) < 0

Inflection Point

f''(x) = 0 and sign changes

Mean Value Theorem

f'(c) = [f(b) - f(a)] / (b - a)

Rolle's Theorem

f'(c) = 0 if f(a) = f(b)