Calculus Refresher: The Ultimate Formula Cheat Sheet

A reference of core formulas and concepts that build on each other — perfect for brushing up before advanced math, physics, or machine learning.

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1️⃣ Limits & Continuity

Limit definition
lim_{x -> a} f(x) = L

Common limit laws

• Sum: lim(f + g) = lim f + lim g
• Product: lim(f * g) = (lim f) * (lim g)
• Quotient: lim(f / g) = (lim f) / (lim g) (denominator ≠ 0)

Special limits

• lim_{x -> 0} (sin x / x) = 1
• lim_{x -> infinity} (1 + 1/x)^x = e

Continuity
f is continuous at x = a if:

• The limit exists and equals the value:
  lim_{x -> a} f(x) = f(a)

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2️⃣ Derivatives (Rates of Change)

Definition
f'(x) = lim_{h -> 0} ( f(x + h) - f(x) ) / h

Power rule
d/dx (x^n) = n * x^(n-1)

Basic rules

• (c)' = 0 (constant)
• (c * f(x))' = c * f'(x)
• (f + g)' = f' + g'
• Product rule: (f * g)' = f' * g + f * g'
• Quotient rule: (f / g)' = (f' * g - f * g') / g^2

Chain rule

• If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x)

Inverse function rule

• If y = f(x) has inverse x = f^{-1}(y), then
  (f^{-1})'(y) = 1 / f'(x) where x = f^{-1}(y)

Common derivatives

• d/dx (x^n) = n x^(n-1)
• d/dx (e^x) = e^x
• d/dx (a^x) = a^x * ln(a)
• d/dx (ln x) = 1 / x (x > 0)
• d/dx (sin x) = cos x
• d/dx (cos x) = -sin x
• d/dx (tan x) = sec^2 x

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3️⃣ Quick Applications of Derivatives

• Critical points: solve f'(x) = 0 or where f' is undefined.
• Increasing / decreasing:
  • f'(x) > 0 ⇒ increasing
  • f'(x) < 0 ⇒ decreasing
• Concavity:
  • f''(x) > 0 ⇒ concave up
  • f''(x) < 0 ⇒ concave down
• Inflection point: f''(x) changes sign.
• Linear approximation (tangent line):
  f(x) ≈ f(a) + f'(a) * (x - a)

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4️⃣ Integrals (Accumulation)

Definition (idea)
∫_a^b f(x) dx = signed area under f(x) from a to b.

Basic rules

• ∫ (k * f(x)) dx = k * ∫ f(x) dx
• ∫ (f(x) + g(x)) dx = ∫ f(x) dx + ∫ g(x) dx

Fundamental Theorem of Calculus

1. If F'(x) = f(x), then
   ∫_a^b f(x) dx = F(b) - F(a)
2. If F(x) = ∫_a^x f(t) dt, then
   F'(x) = f(x)

Power rule for integrals

• ∫ x^n dx = x^(n+1) / (n+1) + C for n ≠ -1
• ∫ (1/x) dx = ln|x| + C

Common indefinite integrals

• ∫ 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
• ∫ sec^2 x dx = tan x + C

Substitution (u-sub)
If u = g(x) and du = g'(x) dx, then
∫ f(g(x)) * g'(x) dx = ∫ f(u) du

Integration by parts

• Choose u and dv so that ∫ u dv is easier as uv - ∫ v du.
  Formula: ∫ u dv = u v - ∫ v du

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5️⃣ A Bit Beyond: Advanced Integration Ideas

Just the labels to jog memory:

• Partial fractions: decompose rational functions into simpler terms before integrating.
• Trig identities: rewrite using
  • sin^2 x + cos^2 x = 1
  • 1 + tan^2 x = sec^2 x, etc.
• Improper integrals: treat infinities or discontinuities as limits.

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6️⃣ Multivariable Calculus

Partial derivatives

• f_x = ∂f/∂x (differentiate w.r.t. x, treat y, z as constants)
• f_y = ∂f/∂y, etc.

Gradient

• For f(x, y, z),
  ∇f = < ∂f/∂x , ∂f/∂y , ∂f/∂z >
  Points in direction of steepest increase.

Directional derivative

• In direction of a unit vector u:
  D_u f = ∇f · u

Double integral

• ∬_R f(x, y) dA = accumulated value of f over region R.
  Often computed as iterated integrals:
  ∫_a^b ∫_c^d f(x, y) dy dx

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7️⃣ Vector Calculus Essentials

Let F = <P, Q, R>.

• Divergence:
  div F = ∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
• Curl:
  curl F = ∇ × F
  (measures local “rotation” of the vector field)

Line integral of a vector field

• If curve C is given by r(t), then
  ∫_C F · dr = ∫ F(r(t)) · r'(t) dt

Surface integral (flux)

• ∬_S F · n dS = total flow of F through surface S
  (n is the unit normal).

Big theorems (just to remember names)

• Green’s Theorem (2D curl ↔ line integral)
• Stokes’ Theorem (curl over surface ↔ line integral along boundary)
• Divergence Theorem (flux through surface ↔ divergence over volume)

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✅ Big Picture

1. Limits → define continuity and derivatives.
2. Derivatives → instantaneous rates of change, slopes, optimization.
3. Integrals → accumulation and area; inverse of derivatives.
4. Multivariable → same ideas in higher dimensions.
5. Vector calculus → connects everything to physics and geometry.

Keep this cheat sheet nearby and you can rebuild most of calculus from these cores.