Home
Calculus
Calculus formula sheet

Calculus formula sheet printable

Q: Does this cover Calculus I and II?

It covers Calculus I fully and the start of Calculus II: basic integrals, u-substitution, integration by parts and Riemann sums.

Q: How do I save it as a PDF?

Click “Print or save as PDF”, then choose “Save as PDF” as the printer.

Q: Why is there a + C on integrals?

An indefinite integral is a family of functions. They differ by a constant, because the derivative of a constant is zero.

Limits, derivative rules, common derivatives and integrals, and applications on one page. Print it or save it as a PDF.

By Chnini Ramdhane · Updated November 2026

Limits

Special limitlim x→0 sin(x)/x = 1

Special limitlim x→0 (1 − cos x)/x = 0

Number elim x→∞ (1 + 1/x)ˣ = e

L'Hôpital's rulelim f(x)/g(x) = lim f′(x)/g′(x)For 0/0 or ∞/∞ forms

Definition of the derivativef′(x) = lim h→0 [f(x + h) − f(x)] / h

Derivative rules

Constantd/dx c = 0

Power ruled/dx xⁿ = n·xⁿ⁻¹

Constant multipled/dx [c·f] = c·f′

Sum rule(f + g)′ = f′ + g′

Product rule(uv)′ = u′v + uv′

Quotient rule(u/v)′ = (u′v − uv′) / v²

Chain ruled/dx f(g(x)) = f′(g(x))·g′(x)

Common derivatives

Sined/dx sin x = cos x

Cosined/dx cos x = −sin x

Tangentd/dx tan x = sec² x

Secantd/dx sec x = sec x tan x

Cosecantd/dx csc x = −csc x cot x

Cotangentd/dx cot x = −csc² x

Exponentiald/dx eˣ = eˣ · d/dx aˣ = aˣ ln a

Logarithmd/dx ln x = 1/x · d/dx logₐ x = 1/(x ln a)

Inverse trigd/dx arcsin x = 1/√(1 − x²) · d/dx arctan x = 1/(1 + x²)

Common integrals

Power∫ xⁿ dx = xⁿ⁺¹/(n + 1) + Cn ≠ −1

Reciprocal∫ 1/x dx = ln|x| + C

Exponential∫ eˣ dx = eˣ + C

Sine∫ sin x dx = −cos x + C

Cosine∫ cos x dx = sin x + C

Secant squared∫ sec² x dx = tan x + C

Inverse tangent∫ 1/(1 + x²) dx = arctan x + C

Integration techniques

Fundamental Theorem of Calculus∫ₐᵇ f(x) dx = F(b) − F(a)F is any antiderivative of f

u-substitution∫ f(g(x))·g′(x) dx = ∫ f(u) duu = g(x)

Integration by parts∫ u dv = uv − ∫ v du

Riemann sum∫ₐᵇ f(x) dx ≈ Σ f(xᵢ)·Δx, Δx = (b − a)/n

Trapezoidal rule(Δx/2)[f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]

Applications

Tangent liney − f(a) = f′(a)(x − a)

Critical pointsf′(x) = 0 or f′(x) undefined

Second derivative testf″(c) > 0 → local min · f″(c) < 0 → local max

Average valuef_avg = 1/(b − a) · ∫ₐᵇ f(x) dx

Area between curvesA = ∫ₐᵇ [f(x) − g(x)] dxf(x) ≥ g(x) on [a, b]

Want practice on your own homework?

Snap any calculus problem with Calculus AI and get a step-by-step solution.

How to use this formula sheet

Use it to check your work, then test yourself by covering the right-hand side. To see a rule in action, type a function into the derivative calculator and follow its steps.

Frequently asked questions

Does this cover Calculus I and II?