Calculus Formula Summary
Key Formulas: Limits · Derivatives · Integrals
1. Limits
1-1. Basic Properties of Limits
Assume f(x) → L and g(x) → M as x → a
Expression
Result
lim [f(x) ± g(x)]
L±M
lim [f(x) · g(x)]
L·M
lim [f(x) / g(x)] (M ≠ 0)
L/M
lim [c · f(x)] (c constant)
c·L
lim [f(x)]ⁿ
Lⁿ
1-2. Common Special Limits
Expression (x→0)
Value
lim sin x / x
1
lim (1 − cos x) / x
0
lim (1 − cos x) / x²
1/2
lim (1 + x)^(1/x)
e
lim (eˣ − 1) / x
1
lim ln(1 + x) / x
1
lim (1 + 1/n)ⁿ (n→∞)
e
1-3. Definition of Continuity
f is continuous at x = a ⇔ ①f(a) is defined, ②lim(x→a) f(x) exists, ③lim(x→a) f(x) = f(a)
2. Differentiation
2-1. Definition of the Derivative
f '(x) = lim(h→0) [f(x+h) − f(x)] / h
2-2. Basic Differentiation Rules
Function
Derivative
c (constant)
0
xⁿ
n·x^(n−1)
c · f(x)
c · f '(x)
f(x) ± g(x)
f '(x) ± g '(x)
Function
Derivative
f(x) · g(x) (Product Rule)
f '(x)g(x) + f(x)g '(x)
f(x) / g(x) (Quotient Rule)
[f '(x)g(x) − f(x)g '(x)] / [g(x)]²
f(g(x)) (Chain Rule)
f '(g(x)) · g '(x)
f⁻¹(x) (Inverse Function)
1 / f '(f⁻¹(x))
2-3. Derivatives of Elementary Functions
Function
Derivative
sin x
cos x
cos x
−sin x
tan x
sec²x
cot x
−csc²x
sec x
sec x·tan x
csc x
−csc x·cot x
eˣ
eˣ
aˣ (a > 0)
aˣ · ln a
ln x
1/x
logₐ x
1 / (x·ln a)
arcsin x
1 / √(1 − x²)
arccos x
−1 / √(1 − x²)
arctan x
1 / (1 + x²)
2-4. Implicit, Parametric & Higher-Order Derivatives
Type
Key Formula
Implicit differentiation
F(x, y) = 0 → differentiate both sides w.r.t. x, then solve for dy/dx
Parametric differentiation
x = x(t), y = y(t) ⇒ dy/dx = (dy/dt) / (dx/dt)
2nd derivative (parametric)
d²y/dx² = d/dt(dy/dx) ÷ (dx/dt)
Logarithmic differentiation
y = f(x)^g(x) → ln y = g(x)·ln f(x), then differentiate both sides
2-5. Mean Value Theorem & Extrema Tests
Theorem / Test
Content
Mean Value Theorem (MVT)
If f is continuous on [a,b] and differentiable on (a,b), then f '(c) =
[f(b)−f(a)]/(b−a) for some c
Rolle's Theorem
If f(a) = f(b), then f '(c) = 0 for some c ∈ (a,b)
First Derivative Test
f ' changes + → − at x=a: local max / − → + : local min
Second Derivative Test
f '(a)=0, f ''(a) > 0 ⇒ local min / f ''(a) < 0 ⇒ local max
Concavity
f ''(x) > 0 : concave up / f ''(x) < 0 : concave down
2-6. L'Hôpital's Rule
For 0/0 or ∞/∞ forms: lim f(x)/g(x) = lim f '(x)/g '(x) (when the latter limit exists)
3. Integration
3-1. Basic Indefinite Integrals
Function
Indefinite Integral (+C)
xⁿ (n ≠ −1)
x^(n+1) / (n+1) + C
1/x
ln|x| + C
eˣ
eˣ + C
aˣ
aˣ / ln a + C
sin x
−cos x + C
cos x
sin x + C
sec²x
tan x + C
csc²x
−cot x + C
sec x·tan x
sec x + C
csc x·cot x
−csc x + C
1 / √(1 − x²)
arcsin x + C
1 / (1 + x²)
arctan x + C
3-2. Properties of Integrals
Property
Statement
Linearity
∫[c·f(x) ± g(x)] dx = c∫f(x)dx ± ∫g(x)dx
Splitting the interval
∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx
Sign reversal
∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx
Fundamental Theorem of
Calculus, Part 1
d/dx ∫ₐˣ f(t)dt = f(x)
Fundamental Theorem of
Calculus, Part 2
∫ₐᵇ f(x)dx = F(b) − F(a) (where F' = f)
3-3. Integration Techniques
Technique
Key Idea / Formula
u-substitution
Let u = g(x): ∫f(g(x))g '(x)dx = ∫f(u)du
Integration by parts
∫u dv = uv − ∫v du (choose u by the LIATE rule)
Trigonometric substitution
√(a²−x²): x=a sinθ / √(a²+x²): x=a tanθ / √(x²−a²): x=a secθ
Partial fractions
Decompose a rational function into terms like (x−a), (x²+bx+c),
then integrate
3-4. Applications of Definite Integrals
Quantity
Formula
Area between two curves
A = ∫ₐᵇ [f(x) − g(x)] dx (f ≥ g)
Volume (disk method)
V = π∫ₐᵇ [f(x)]² dx
Volume (shell method)
V = 2π∫ₐᵇ x·f(x) dx
Quantity
Formula
Arc length
L = ∫ₐᵇ √(1 + [f '(x)]²) dx
Surface area of revolution
S = 2π∫ₐᵇ f(x)√(1 + [f '(x)]²) dx
Average value (integral)
f_avg = 1/(b−a) · ∫ₐᵇ f(x) dx
4. Sequences and Series
4-1. Standard Series
Series
Sum
Geometric series Σ arⁿ (|r| < 1,
n=0→∞)
a / (1 − r)
p-series Σ 1/nᵖ
Converges if p > 1, diverges if p ≤ 1
Harmonic series Σ 1/n
Diverges
4-2. Convergence Tests
• Ratio Test: lim|aₙ₊₁/aₙ| = L → converges if L<1, diverges if L>1, inconclusive if L=1
• Root Test: lim ⁿ√|aₙ| = L → converges if L<1, diverges if L>1, inconclusive if L=1
• Integral Test: if f(x) is positive, decreasing, and continuous, Σf(n) and ∫₁^∞ f(x)dx share the same convergence
behavior
• Alternating Series Test: if aₙ is decreasing and aₙ→0, then Σ(−1)ⁿaₙ converges
4-3. Taylor / Maclaurin Series
f(x) = Σ [fⁿ(a) / n!] (x − a)ⁿ (the Maclaurin series is the case a = 0)
Function
Maclaurin Expansion
eˣ
1 + x + x²/2! + x³/3! + ⋯
sin x
x − x³/3! + x⁵/5! − ⋯
cos x
1 − x²/2! + x⁴/4! − ⋯
ln(1 + x)
x − x²/2 + x³/3 − ⋯ (|x| < 1)
1 / (1 − x)
1 + x + x² + x³ + ⋯ (|x| < 1)
5. Multivariable Calculus Basics
5-1. Partial Derivatives
∂f/∂x: differentiate with respect to x, treating y as constant / ∂f/∂y: differentiate with respect to y, treating x as constant
5-2. Total Differential
df = (∂f/∂x)dx + (∂f/∂y)dy
Note: This is a condensed formula reference for exam preparation. Refer to your course textbook for full derivations and proofs.