Formulas & Equation Cheat-Sheets

Fundamental definition of derivative as instantaneous rate of change

The derivative of any constant c is zero

Valid for all real exponents n (n ∈ ℝ)

Constants factor out of differentiation

Differentiation is a linear operator

Differentiate first, hold second, plus hold first, differentiate second

Requires g(x) ≠ 0

Differentiate outer function evaluated at inner, multiply by derivative of inner

Direct shortcut for reciprocating functions

The unique non-zero function equal to its own derivative

Chain rule applied to natural exponential

Valid for a > 0, a ≠ 1

For absolute values: d/dx [ln|x|] = 1 / x (x ≠ 0)

Logarithmic derivative / relative rate of change

Change of base differentiation formula

Derived using logarithmic differentiation y = xˣ ⇒ ln y = x ln x

Rate of change of sine is cosine

Notice the negative sign for "co-" function

Also equal to 1 + tan²(x)

Also equal to -(1 + cot²(x))

Product of secant and tangent

Product of cosecant and cotangent with negative sign

Standard generalized trigonometric chain rule

Domain strictly between -1 and 1

Exact negative of inverse sine derivative

Valid for all real numbers without singularity

Exact negative of inverse tangent derivative

Requires |x| > 1

Exact negative of inverse secant derivative

Notice no negative sign (unlike regular cosine)

Positive rate of change

Crucial for neural network activation backpropagation

Negative hyperbolic cosecant squared

Negative hyperbolic product

Negative hyperbolic product

Valid for all real x

Valid for x > 1

Valid for |x| < 1

Calculates curve tangent slope from parametric equations x(t), y(t)

Calculates concavity of parametric curves

Resolves indeterminate limits by differentiating numerator and denominator independently

Guarantees a point where instantaneous rate of change equals average rate of change

Polynomial approximation of smooth differentiable functions

Taylor series centered at the origin (x = 0)

Antiderivative of a constant k is k·x + C

Valid for all real powers n except n = -1

Crucial special case for n = -1 in power rule

Integration is a linear functional operation

Shortcut for linear binomial powers

Standard linear denominator integration formula

Function equals its own antiderivative up to constant C

Standard scaled exponential integration rule

Divides by ln(a) because derivative multiplied by ln(a)

Derived using integration by parts with u = ln x, dv = dx

Change of base integration for logarithms

Classic integration by parts application

Note the negative sign in antiderivative of sine

Direct standard trigonometric antiderivative

Because derivative of tangent is secant squared

Because derivative of cotangent is -csc²(x)

Direct derivative reversal of sec(x)

Direct derivative reversal of csc(x)

Evaluated by substitution u = cos(x)

Evaluated by substitution u = sin(x)

Famous calculus integral multiplied by (sec x + tan x)/(sec x + tan x)

Standard logarithmic cosecant integral

Evaluated using power reduction identity sin²(x) = (1 - cos 2x)/2

Evaluated using power reduction identity cos²(x) = (1 + cos 2x)/2

Evaluated using identity tan²(x) = sec²(x) - 1

Evaluated using identity cot²(x) = csc²(x) - 1

Notice no 1/a factor outside for inverse sine

The 1/a coefficient is required unlike inverse sine

Includes 1/a coefficient and absolute value |x|

Obtained via partial fraction decomposition 1/(x²-a²)

Obtained via partial fraction decomposition 1/(a²-x²)

Used for computing circular and elliptical sector areas

Solved using substitution x = a tan θ or hyperbolic x = a sinh u

Solved using substitution x = a sec θ or hyperbolic x = a cosh u

Also equal to arsinh(x/a) and arcosh(x/a) respectively

Positive cosh(x) + C (no negative sign)

Direct hyperbolic antiderivative

Since cosh x > 0 for all real x, no absolute value required

Direct antiderivative yielding tanh(x)

Negative hyperbolic cotangent

Antiderivative of hyperbolic secant-tangent product

Choose u using LIATE rule (Logarithmic, Inverse trig, Algebraic, Trig, Exponential)

Reverses the chain rule of differentiation

Transforms any rational trigonometric integral into a rational polynomial algebraic integral

Whenever numerator is exact derivative of denominator

Connects differentiation and integration as inverse mathematical operations

Evaluates net area and accumulated change over interval [a, b]

Chain rule extended to variable lower and upper integration limits

Instantaneous simplification for symmetric integration boundaries [-a, a]

Calculates the true average value of a continuous function over [a, b]

Reduces power of sine by 2 on each iteration

Reduces power of cosine by 2 on each iteration

Fast algebraic reduction for tanⁿ(x)

Reduces higher powers of secant (e.g. ∫sec³ x dx)

Solutions to standard quadratic equation ax² + bx + c = 0; discriminant Δ = b² - 4ac determines real/complex nature

Fundamental polynomial expansions of algebraic binomials

Expansion of square of a sum of three terms

Exact algebraic factorizations for sum and difference of cubic terms

Special case: if a + b + c = 0, then a³ + b³ + c³ = 3abc

Expands any positive integer power of a binomial sum using combinatorial coefficients

Direct relationships between polynomial roots and coefficients

Equality holds if and only if all elements a₁ = a₂ = ... = a_n

Universal inequality in real/complex inner product spaces

Straight-line distance and internal/external ratio division of line segment connecting (x₁, y₁) and (x₂, y₂)

Calculates planar area from coordinates of 3 vertices; zero area implies collinearity

Point-slope equation and shortest perpendicular distance from point (x₀, y₀) to line Ax + By + C = 0

Acute intersection angle; parallel lines have m₁ = m₂; perpendicular lines have m₁·m₂ = -1

Circle equation with center (h, k) and radius r

Locus of points equidistant from focus and directrix (eccentricity e = 1)

Standard horizontal ellipse with semi-major axis a, semi-minor axis b, and eccentricity e < 1

Conic section with eccentricity e > 1 and linear asymptotes passing through origin

Euclidean distance in 3D cartesian space and sphere equation with center (x₀, y₀, z₀) and radius R

Volumetric capacity and total surface area formulas for solid 3D spheres and cylinders

Slant height l, lateral surface area A_lat = π·r·l, and truncated cone frustum volume

Solid of revolution generated by revolving a circle of radius r around an axis at major radius R from center (R ≥ r)

Equation of plane passing through point (x₀, y₀, z₀) with normal vector n = <A, B, C>

Calculates absolute shortest distance between two non-intersecting, non-parallel 3D lines r₁ = a₁ + λb₁ and r₂ = a₂ + μb₂

Scalar value scaling volume under linear transformation; det(A) = 0 indicates non-invertible singular matrix

Matrix inverse where adj(A) is the transpose of the cofactor matrix C

Explicit algebraic formula for solving system of n linear equations Ax = b where A_i has column i replaced by b

Characteristic polynomial roots yield scalar eigenvalues λ; corresponding non-zero vectors v are eigenvectors

Matrix trace equals sum of eigenvalues; determinant equals product of all eigenvalues

Fundamental vector products: dot product yields scalar projection; cross product yields orthogonal vector

Scalar triple product computes volume of parallelepiped; BAC-CAB rule expands double cross products

Vector pointing in the direction of maximum rate of increase of scalar function f; magnitude equals maximum slope

Divergence measures outward flux density (∇·F = 0 is solenoidal); Curl measures rotational vorticity (∇×F = 0 is irrotational)

Second-order differential operator governing harmonic potentials, heat diffusion, and wave propagation

Relates counterclockwise line integral around positively oriented simple closed curve C to double integral over enclosed region D

Relates circulation line integral along boundary curve C to surface flux integral of curl ∇×F over orientable surface S

Outward net flux of vector field F across closed surface S equals volume integral of divergence ∇·F throughout interior volume V

Area/volume scaling factor during multivariable change of variables (Polar: J = r; Spherical: J = ρ² sin φ)

Geometric representation of complex number in Argand plane

Computes integer/fractional powers and symmetric n-th roots in the complex plane

Necessary and sufficient conditions for complex function f(z) = u(x, y) + i·v(x, y) to be holomorphic / complex-differentiable

Evaluates contour integrals via enclosed singularities and residues Res(f, z₀) = lim[z→z₀] (z - z₀)f(z)

Sum of geometric progression with first term a and common ratio r

Representation of smooth function as infinite polynomial sum of derivatives at expansion point a (Maclaurin when a = 0)

Infinite power series representations with infinite radius of convergence (R = ∞)

Decomposes periodic waveform of period 2L into infinite sum of harmonic sinusoidal orthogonal components

General solution method using integrating factor I(x) = exp(∫P dx)

Roots determine solution form: distinct real (C₁e^λ₁x + C₂e^λ₂x), repeated ((C₁ + C₂x)e^λx), complex α±iβ (e^αx(C₁cos βx + C₂sin βx))

Transforms linear differential equations into algebraic equations in complex frequency s-domain

Three fundamental linear second-order partial differential equations governing physics and engineering

Calculates posterior probability P(A|B) given prior probability P(A) and conditional likelihood P(B|A)

First and second central moments measuring mean center and dispersion around the mean

Probability of exactly k successes in n independent Bernoulli trials with success probability p

Models rare discrete event occurrences in fixed interval of time/space with constant average rate λ

Continuous symmetric bell-shaped probability density function governing Central Limit Theorem

Ratio of total displacement to the total time interval taken

First time derivative of position coordinate vector with respect to time

Rate of change of velocity with respect to time or position

Relates final velocity to initial velocity under uniform acceleration a

Computes total linear displacement under constant acceleration

Time-independent kinematic relation for constant acceleration

Calculates displacement during specifically the n-th second of motion

Valid for motion under uniform constant linear acceleration

Peak vertical altitude reached by a projectile launched at angle θ

Total airborne time from launch to landing on level ground

Maximum horizontal distance traveled (maximum at launch angle θ = 45°)

Cartesian path coordinates (x, y) demonstrating pure parabolic motion

Velocity of body A observed from the reference frame of moving body B

Product of mass and velocity; conserved in isolated systems with zero external force

Net external force equals the instantaneous rate of change of linear momentum

The impulse of a force over duration Δt equals the total change in momentum produced

Self-adjusting resistive force opposing the tendency of relative motion up to threshold

Constant resistive force opposing active relative sliding motion (typically μ_k < μ_s)

Minimum incline tilt angle at which a body begins to slide down under gravity

Inward radially directed acceleration for circular trajectory of radius r

Net inward force required to maintain circular trajectory motion

Speed at which a vehicle negotiates a turn with zero side friction wear

Upper speed limit before outward skidding occurs on banked surface with friction

Period of bob describing horizontal circle of cone angle θ and string length L

Minimum threshold speeds to complete a vertical loop without string slackening

Scalar dot product of force vector and displacement vector

Area under the Force-Displacement graph between initial and final positions

Net work done by all forces (conservative, non-conservative, external) equals change in kinetic energy

Energy possessed by a body due to its state of motion

Potential energy stored in a mass m raised to vertical height h in uniform field g

Energy stored when a spring of stiffness constant k is stretched/compressed by x

Conservative forces always point in the direction of steepest potential energy decrease

Time rate at which mechanical work is performed or energy transferred

Measures elasticity: e=1 (kinetic energy conserved), e=0 (objects stick together)

Exact post-collision speeds for two masses colliding elastically in 1 dimension

Kinetic energy converted into internal thermal deformation and acoustic energy

Mean position of mass distribution for discrete or continuous solid bodies