This page collects every major formula from a first-year calculus course in one place. Use it for quick lookup during practice — not as a substitute for understanding where the formulas come from. If a formula looks unfamiliar, follow the link to the relevant article.
Limit Formulas
All Derivative Rules
Integration Formulas
Fundamental Theorems
Key Taylor/Maclaurin Series
Convergence Tests for Series
This cheat sheet covers the formulas most commonly needed in first-year calculus. For each formula: know what it computes, what conditions are required, and what a typical application looks like. Memorising without understanding is fragile under exam pressure.
Limit Formulas and Theorems
Formal Definition: lim(x→a) f(x) = L means: for every ε > 0, there exists δ > 0 such that 0 < |x−a| < δ implies |f(x)−L| < ε.
L'Hôpital's Rule: If lim f/g gives 0/0 or ∞/∞, then lim f/g = lim f'/g' (conditions: differentiable near a, lim g' ≠ 0).
Squeeze Theorem: If g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L.
Standard trig limits: lim(x→0) sin(x)/x = 1. lim(x→0) (1−cos x)/x = 0. lim(x→0) (1−cos x)/x² = 1/2.
Key limit values: lim(x→0) (eˣ−1)/x = 1. lim(x→0) ln(1+x)/x = 1. lim(x→∞)(1+1/n)ⁿ = e.
Derivative Rules — Complete Reference
Basic rules: d/dx[c] = 0. d/dx[xⁿ] = nxⁿ⁻¹. d/dx[cf(x)] = c·f'(x). (f±g)' = f'±g'.
Product Rule: (fg)' = f'g + fg'.
Quotient Rule: (f/g)' = (f'g − fg')/g².
Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x). Leibniz form: dy/dx = (dy/du)·(du/dx).
Exponential/Log: d/dx[eˣ]=eˣ. d/dx[aˣ]=aˣln(a). d/dx[ln x]=1/x. d/dx[log_a x]=1/(x·ln a).
Trig: (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.
Inverse trig: (arcsin x)'=1/√(1−x²). (arccos x)'=−1/√(1−x²). (arctan x)'=1/(1+x²). (arccot x)'=−1/(1+x²).
Integration Formulas — Complete Reference
Power: ∫xⁿ dx = xⁿ⁺¹/(n+1)+C (n≠−1). ∫x⁻¹ dx = ln|x|+C.
Exponential/log: ∫eˣ dx = eˣ+C. ∫aˣ dx = aˣ/ln(a)+C. ∫ln x dx = x·ln x − x+C.
Trig: ∫sin x dx=−cos x+C. ∫cos x dx=sin x+C. ∫tan x dx=ln|sec x|+C. ∫sec x dx=ln|sec x+tan x|+C. ∫sec²x dx=tan x+C. ∫csc²x dx=−cot x+C.
Inverse trig: ∫1/√(1−x²) dx=arcsin x+C. ∫1/(1+x²) dx=arctan x+C. ∫1/√(a²−x²) dx=arcsin(x/a)+C. ∫1/(a²+x²) dx=(1/a)arctan(x/a)+C.
Techniques: U-substitution: u=g(x), du=g'(x)dx → ∫f(g(x))g'(x)dx = ∫f(u)du. IBP: ∫u dv = uv − ∫v du (LIATE for u).
Key Theorems
IVT: f continuous on [a,b], y₀ between f(a) and f(b) → ∃c∈(a,b): f(c)=y₀.
EVT: f continuous on [a,b] → f attains absolute max and min on [a,b].
MVT: f continuous on [a,b], differentiable on (a,b) → ∃c: f'(c) = (f(b)−f(a))/(b−a).
Rolle's Theorem: f(a)=f(b) → ∃c: f'(c)=0 (special case of MVT).
FTC Part 1: d/dx[∫ₐˣ f(t)dt] = f(x). With upper limit g(x): multiply result by g'(x).
FTC Part 2: ∫ₐᵇ f(x)dx = F(b)−F(a) for any antiderivative F.
Series (BC / University Level)
Geometric: Σarⁿ = a/(1−r) for |r|<1. Diverges for |r|≥1.
p-series: Σ1/nᵖ converges iff p>1. Σ1/n (harmonic series) diverges.
Ratio Test: L = lim|aₙ₊₁/aₙ|. L<1: converges. L>1: diverges. L=1: inconclusive.
Alternating Series Test: Σ(−1)ⁿbₙ converges if bₙ decreasing → 0.
Key Maclaurin Series: eˣ = Σxⁿ/n! (all x). sin x = Σ(−1)ⁿx^(2n+1)/(2n+1)! (all x). cos x = Σ(−1)ⁿx^(2n)/(2n)! (all x). 1/(1−x) = Σxⁿ (|x|<1). ln(1+x) = Σ(−1)ⁿ⁺¹xⁿ/n (|x|≤1, x≠−1).
- Abramowitz, M. & Stegun, I. (1972). Handbook of Mathematical Functions. Dover.
- Stewart, J. (2015). Calculus, 8th ed., inside covers. Cengage.
- Gradshteyn, I.S. & Ryzhik, I.M. (2007). Table of Integrals, Series and Products. Academic Press.