Derivative definition
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$
Define instantaneous rate of change as a limit of secant slopes.
AP · UniversityCalculus formulas, grouped by topic.
42 calculus formulas across 12 topics, free and without an account. Calculus formulas describe change and accumulation. Derivative rules break a complicated function into pieces you already know how to differentiate, integral rules reverse that process, and the fundamental theorem of calculus ties the two directions together. The application formulas — area, volume, arc length, average value — are all the same move: write the small piece, then integrate it over the interval.
Every calculus formula on one page
Each entry links to its own page, where the symbols are defined, the conditions are stated, and the formula is substituted through a worked example. Levels covered here: High school, AP, University.
Limits
2 formulas in limits.
L’Hôpital’s rule
$$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$$
Evaluate certain indeterminate quotients by differentiating numerator and denominator separately.
AP · UniversityDerivatives
9 formulas in derivatives.
Power rule
$$\frac{d}{dx}x^n=nx^{n-1}$$
Differentiate a power of the variable.
AP · UniversityProduct rule
$$(fg)'=f'g+fg'$$
Differentiate a product without treating it as two separate derivatives.
AP · UniversityQuotient rule
$$\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}$$
Differentiate a quotient of two functions.
AP · UniversityChain rule
$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$
Differentiate a composition from the outside inward.
AP · UniversityExponential and logarithmic derivatives
$$\frac{d}{dx}e^x=e^x,\quad\frac{d}{dx}a^x=a^x\ln a,\quad\frac{d}{dx}\ln x=\frac1x$$
Differentiate the fundamental exponential and logarithm functions.
AP · UniversityTrigonometric derivatives
$$(\sin x)'=\cos x,\quad(\cos x)'=-\sin x,\quad(\tan x)'=\sec^2x$$
Differentiate sine, cosine, and tangent.
AP · UniversityInverse trigonometric derivatives
$$(\arcsin x)'=\frac1{\sqrt{1-x^2}},\quad(\arctan x)'=\frac1{1+x^2}$$
Differentiate two common inverse trigonometric functions.
AP · UniversityRemaining trigonometric derivatives
$$(\cot x)'=-\csc^2x,\quad(\sec x)'=\sec x\tan x,\quad(\csc x)'=-\csc x\cot x$$
Differentiate cotangent, secant, and cosecant.
AP · UniversityRemaining inverse-trig derivatives
$$(\arccos x)'=-\frac1{\sqrt{1-x^2}},\quad(\operatorname{arccot}x)'=-\frac1{1+x^2}$$
Differentiate inverse cosine and the common decreasing convention for inverse cotangent.
AP · UniversityApproximation
1 formula in approximation.
Linearization
$$L(x)=f(a)+f'(a)(x-a)$$
Approximate a differentiable function near a known point.
AP · UniversityIntegration
7 formulas in integration.
Power antiderivative
$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$$
Reverse the power rule to find an antiderivative.
AP · UniversitySubstitution rule
$$\int f(g(x))g'(x)\,dx=\int f(u)\,du$$
Reverse the chain rule by replacing an inner expression.
AP · UniversityIntegration by parts
$$\int u\,dv=uv-\int v\,du$$
Reverse the product rule to integrate a product.
AP · UniversityFundamental theorem of calculus
$$\int_a^b f(x)\,dx=F(b)-F(a),\qquad F'=f$$
Evaluate signed accumulated change using any antiderivative.
AP · UniversityAccumulation derivative
$$\frac{d}{dx}\int_a^x f(t)\,dt=f(x)$$
Differentiate an accumulation function with a variable upper bound.
AP · UniversityCommon exponential and trigonometric antiderivatives
$$\int e^x dx=e^x+C,\quad\int\cos x\,dx=\sin x+C,\quad\int\sin x\,dx=-\cos x+C$$
Recall three fundamental antiderivatives.
AP · UniversityImproper integral definition
$$\int_a^{\infty}f(x)dx=\lim_{b\to\infty}\int_a^b f(x)dx$$
Define an integral over an unbounded interval using a limit.
UniversityApplications
7 formulas in applications.
Average value of a function
$$f_{\rm avg}=\frac1{b-a}\int_a^b f(x)\,dx$$
Find the constant height with the same signed area on an interval.
AP · UniversityArea between curves
$$A=\int_a^b|f(x)-g(x)|\,dx$$
Add vertical distances between two curves.
AP · UniversityArc length of a graph
$$L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx$$
Add infinitesimal straight-line lengths along $y=f(x)$.
AP · UniversityDisk and washer volumes
$$V=\pi\int_a^b\left(R(x)^2-r(x)^2\right)dx$$
Build a solid of revolution from circular cross-sections.
AP · UniversityCylindrical shell volume
$$V=2\pi\int_a^b(\text{radius})(\text{height})\,dx$$
Build a solid of revolution from thin cylindrical shells.
AP · UniversityPosition, velocity, and acceleration
$$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t)$$
Relate an object’s position to instantaneous velocity and acceleration.
High school · APSurface area of revolution
$$S=2\pi\int_a^b r(x)\sqrt{1+[f'(x)]^2}\,dx$$
Add the curved bands formed by revolving a graph around an axis.
AP · UniversitySeries
6 formulas in series.
Taylor series
$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$$
Represent a sufficiently regular function by derivatives at a center.
AP · UniversityTaylor remainder bound
$$|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}$$
Bound the error after truncating a Taylor polynomial.
AP · UniversityCommon Maclaurin series
$$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!},\quad\sin x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!},\quad\cos x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}$$
Use standard power series for exponential, sine, and cosine.
AP · UniversityGeometric series test
$$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}\quad\text{when }|r|<1$$
Determine convergence and sum of an infinite geometric series.
AP · UniversityRatio test
$$L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$$
Test absolute convergence by comparing successive term sizes.
AP · UniversityAlternating-series error bound
$$|R_n|\le b_{n+1}$$
Bound the error after stopping a convergent alternating series.
AP · UniversityNumerical methods
1 formula in numerical methods.
Newton’s method
$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$
Improve an estimate of a root by following the tangent line to the axis.
AP · UniversityParametric
2 formulas in parametric.
Parametric derivatives
$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\qquad\frac{d^2y}{dx^2}=\frac{d}{dt}(dy/dx)\Big/\frac{dx}{dt}$$
Differentiate a curve whose coordinates depend on a parameter.
AP · UniversityParametric arc length
$$L=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$$
Measure length along a parametrized plane curve.
AP · UniversityPolar
2 formulas in polar.
Polar area
$$A=\frac12\int_{\alpha}^{\beta}r(\theta)^2\,d\theta$$
Find area swept by a polar curve.
AP · UniversityPolar arc length
$$L=\int_{\alpha}^{\beta}\sqrt{r(\theta)^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta$$
Measure length along a polar curve.
AP · UniversityMultivariable
3 formulas in multivariable.
Gradient
$$\nabla f=\left\langle f_x,f_y,f_z\right\rangle$$
Collect partial derivatives into the direction of steepest increase.
UniversityDirectional derivative
$$D_{\mathbf u}f=\nabla f\cdot\mathbf u$$
Measure change per unit distance in a chosen direction.
UniversityDouble and triple integrals
$$\iint_R f\,dA,\qquad\iiint_E f\,dV$$
Accumulate a density or function across a region or solid.
UniversityVector calculus
1 formula in vector calculus.
Line integral of a vector field
$$\int_C\mathbf F\cdot d\mathbf r=\int_a^b\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)dt$$
Accumulate the tangential component of a vector field along a curve.
UniversityTheorems
1 formula in theorems.
Mean value theorem
$$f'(c)=\frac{f(b)-f(a)}{b-a}\quad\text{for some }c\in(a,b)$$
Guarantee a tangent slope equal to the interval’s average rate of change.
AP · UniversityKeep going with calculus
A formula sheet is a reference, not a method. Use the subject guide to see where these relationships come from, a calculator to check a substitution you are unsure about, and practice questions to find out whether you can actually retrieve them.
- Study calculus — the subject hub with lessons, worked methods, and common mistakes.
- Calculus calculator — enter your own values and compare the result with your working.
- Calculus Practice — retrieval practice across every topic listed above.
- Math formula library — search all 215 formulas by name, symbol, or condition.
Formulas in the other subjects
Mathematics does not stop at a subject boundary, and neither do its formulas. Each subject page below lists its own relationships, conditions, and worked examples.
Arithmetic
Review the core arithmetic relationships, their restrictions, and worked substitutions.
Algebra
Review the core algebra relationships, their restrictions, and worked substitutions.
Geometry
Review the core geometry relationships, their restrictions, and worked substitutions.
Trigonometry
Review the core trigonometry relationships, their restrictions, and worked substitutions.
Precalculus
Review the core precalculus relationships, their restrictions, and worked substitutions.
Statistics
Review the core statistics relationships, their restrictions, and worked substitutions.
Probability
Review the core probability relationships, their restrictions, and worked substitutions.
Linear Algebra
Review the core linear algebra relationships, their restrictions, and worked substitutions.
Differential Equations
Review the core differential equations relationships, their restrictions, and worked substitutions.
Discrete Math
Review the core discrete math relationships, their restrictions, and worked substitutions.