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Calculus basics cheat sheet

All 12 formulas on one page. Print it or save it as a PDF; it fits on 2 sides of A4 or Letter.

Power rule for derivatives

d/(dx)(axⁿ) = nax^(n − 1)

Constant

d/(dx)(c) = 0

Sum rule

d/(dx)(f + g) = f′ + g′

$m$ the slope (gradient) of the curve at the chosen x: the value of $f′(x)$
$a$ the coefficient in $f(x) = axⁿ$
$n$ the power (any real number)
$x$ the point where you want the slope

Product rule

(uv)′ = u′v + uv′

$D$ the value of the derivative $(uv)′$ at the chosen x
$m$ in the calculator, the power in $u = x^m$
$k$ in the calculator, the rate in $v = e^(kx)$
$x$ the point where the derivative is evaluated
$u$ the first factor
$v$ the second factor
$u'$ the derivative of u

Quotient rule

(u/v)′ = (u′v − uv′)/(v²)

$D$ the value of the derivative $(u/v)′$ at the chosen x
$a$ in the calculator, $u = ax + b$
$b$ the constant in u
$c$ in the calculator, $v = cx + d$
$d$ the constant in v
$x$ the point where the derivative is evaluated
$v$ the denominator (it must not be 0)

Chain rule

dy/(dx) = dy/(du)·du/(dx)

Bracket form

d/(dx)(ax + b)ⁿ = na(ax + b)^(n − 1)

Function notation

(f(g(x)))′ = f′(g(x)) g′(x)

$D$ the value of $dy/dx$ at the chosen x
$a$ in the calculator, the coefficient of x inside $(ax + b)ⁿ$
$b$ the constant inside the bracket
$n$ the outer power
$x$ the point
$u$ the inside function
$y$ the whole function, written in terms of u

Derivatives of sin and cos

d/(dx)sin (kx) = kcos (kx), d/(dx)cos (kx) = −ksin (kx)

Tangent

d/(dx)tan x = 1/(cos ²x)

$D$ the value of $f′(x)$ at the chosen x
$A$ the coefficient of $sin (kx)$
$B$ the coefficient of $cos (kx)$
$k$ the number multiplying x inside (angular frequency)
$x$ the point, in radians

Derivatives of eˣ and ln x

d/(dx)e^(kx) = ke^(kx), d/(dx)ln x = 1/x

Other bases

d/(dx)a^x = a^xln a

Log of a function

d/(dx)ln (g(x)) = g′(x)/(g(x))

$D$ the value of $f′(x)$ at the chosen x
$A$ the coefficient of $e^(kx)$
$k$ the rate in $e^(kx)$
$B$ the coefficient of $\ln x$
$x$ the point (must be positive if there is a ln term)

Power rule for integrals

∫xⁿ dx = (x^(n + 1))/(n + 1) + C, n ≠ − 1

Definite form (the calculator)

∫ₐ^bcxⁿ dx = c (b^(n + 1) − a^(n + 1))/(n + 1)

The n = −1 case

∫1/x dx = ln |x| + C

$I$ the value of the definite integral $∫ₐ^bcxⁿ dx$
$c$ the coefficient
$n$ the power (not −1)
$a$ the lower limit
$b$ the upper limit
$C$ the constant of integration (indefinite integrals only)

Integral of eᵏˣ

∫e^(kx) dx = 1/ke^(kx) + C

$I$ the value of $∫ₐ^bAe^(kx) dx$
$A$ the coefficient
$k$ the rate (if 0, the integrand is constant)
$a$ the lower limit
$b$ the upper limit
$C$ the constant of integration

Definite integral (area under a curve)

∫ₐ^bf(x) dx = F(b) − F(a)

$I$ the value of the integral: the signed area
$c_{3}$ the coefficient of $x^{3}$ in f
$c_{2}$ the coefficient of $x^{2}$
$c_{1}$ the coefficient of x
$c_{0}$ the constant term
$a$ the lower limit
$b$ the upper limit
$F$ an antiderivative of f (so $F′ = f$ )

Average value of a function

f̄ = 1/(b − a)∫ₐ^bf(x) dx

$\bar{f}$ the average value of f on the interval
$c_{3}$ the coefficient of $x^{3}$ in f
$c_{2}$ the coefficient of $x^{2}$
$c_{1}$ the coefficient of x
$c_{0}$ the constant term
$a$ the start of the interval
$b$ the end of the interval (not equal to a)

Equation of a tangent line

y = f(x₀) + f′(x₀)(x − x₀)

Normal line

y = f(x₀) − 1/(f′(x₀))(x − x₀)

$m$ the slope of the tangent, $f′(x₀)$
$c_{3}$ the coefficient of $x^{3}$ in f
$c_{2}$ the coefficient of $x^{2}$
$c_{1}$ the coefficient of x
$c_{0}$ the constant term
$x_{0}$ the x-coordinate of the point of contact

Difference quotient

f′(x) = lim[h → 0] (f(x + h) − f(x))/h

$Q$ the difference quotient: the slope of the secant through x and x + h
$c_{3}$ the coefficient of $x^{3}$ in f
$c_{2}$ the coefficient of $x^{2}$
$c_{1}$ the coefficient of x
$c_{0}$ the constant term
$x$ the point
$h$ the step (any non-zero number)

letx.io · Calculus basics cheat sheet · 2026-10-31