Do all functions have an antiderivative I can write down?

No. Some simple-looking functions, such as e^(−x²) and sin(x)/x, have antiderivatives that cannot be written using elementary functions. Their definite integrals are still computed accurately with numerical methods or special functions.

What is L'Hôpital's rule used for?

It evaluates limits that take the indeterminate forms 0/0 or ∞/∞. If both the numerator and denominator approach 0 or both approach infinity, the limit equals the limit of the ratio of their derivatives, provided that limit exists. It sometimes has to be applied more than once.

How accurate are numerical methods like Simpson's rule or RK4?

For smooth functions they are very accurate: halving the step size cuts the error of Simpson's rule by about a factor of 16 and of RK4 by about the same. Accuracy drops sharply near discontinuities, singularities or rapidly changing behavior, so check results with a smaller step.

Why do Taylor series only work near a point?

A Taylor series is built from derivatives at a single point, so it describes the function best nearby. Many series converge only within a certain radius, and outside it the partial sums move away from the true value. A few functions, such as e^x, sin x and cos x, have series that converge everywhere.

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Calculus calculators

These tools cover differential and integral calculus and the methods built on them: derivatives and tangent lines, definite, improper and numerical integrals, limits, series, Fourier and Laplace transforms, and ordinary differential equations. Several are numerical methods, which approximate an answer when an exact formula is unavailable or impractical.

21 tools · Updated October 21, 2026

Showing 21 tools

∫Arc Length CalculatorIntegrals and their applicationsArc length of y = x² from 0 to 24.646784

∫Area Between Curves CalculatorIntegrals and their applicationsArea between sin x and cos x on [0, π]2.828427

d/dxDerivative CalculatorDerivativesd/dx (x³ − 3x)3x² − 3

y′Differential Equation SolverDifferential equations and transformsy′ = y − x² + 1, y(0) = 0.5, find y(2)y(2) ≈ 5.305363

∫Fourier Series CalculatorLimits and seriesFourier series of sign(x) on [−π, π]1.27324sin(x) + …

∫Improper Integral CalculatorIntegrals and their applications∫ e^(−x²) dx from −∞ to ∞1.77245385

∫Integral CalculatorIntegrals and their applications∫ (x³ − 4x) dx from −1 to 34

∫Interpolation CalculatorRoot finding and interpolationBessel J₀ table (5 points) at x = 1.5P(1.5) = 0.51182

ℒLaplace Transform CalculatorDifferential equations and transformsLaplace transform of e^(−t)·sin(3t)3/((s + 1)² + 9)

limLimit CalculatorLimits and serieslim x→0 of sin(x)/x1

∫Newton's Method CalculatorRoot finding and interpolationRoot of cos(x) − x from x₀ = π/4x ≈ 0.739085133215

f′Numerical Differentiation CalculatorDerivativesf′(1.8) for ln(x), h = 0.010.5555612713

∫Numerical Integration CalculatorIntegrals and their applications∫ e^(−x²) dx, 0 to 2, Simpson n = 80.8820655104

∂Partial Derivative CalculatorDerivatives∇f for f = x²y + y³ at (1, 2)(4, 13)

ΣRiemann Sum CalculatorIntegrals and their applicationsLeft sum of x² on [0, 2], n = 82.1875

f″Second Derivative CalculatorDerivativesf″(1) for f(x) = x³ − 6x²−6, concave down

ΣSeries Convergence CalculatorLimits and seriesΣ 1/n² from n = 1 to ∞Converges absolutely

∫Tangent Line CalculatorDerivativesTangent to y = x³ − 3x at x = 2y = 9x − 16

ΣTaylor Series CalculatorLimits and seriesOrder 6 Maclaurin for eˣ at x = 1P(1) = 2.71805556

∫Volume of Revolution CalculatorIntegrals and their applicationsy = √x, 0 to 4, spun about the x-axis8π ≈ 25.132741

WWronskian CalculatorDifferential equations and transformsW(eˣ, e²ˣ) at x = 01, independent

Which one do I need?

I need the derivative of a function, with the rules shownDerivative Calculator I want the equation of the tangent line at a pointTangent Line Calculator I need a definite integral evaluatedIntegral Calculator I need to integrate up to infinity or across a point where the function blows upImproper Integral Calculator I want to evaluate a limit, including 0/0 and ∞/∞ formsLimit Calculator I want a polynomial approximation of a function near a pointTaylor Series Calculator I need to find a root numerically from a starting guessNewton's Method Calculator

Derivatives: rates of change and slopes

A derivative measures how fast a function changes at a point, which is the slope of its tangent line there. The Derivative Calculator applies the power, product, quotient and chain rules and shows which rule was used at each step. The Second Derivative Calculator tells you about concavity and helps classify maxima and minima, and the Partial Derivative Calculator handles functions of several variables.

When a function is known only as data or is expensive to differentiate symbolically, finite differences give an approximation. The Numerical Differentiation Calculator compares forward, central and Richardson estimates so you can see how the error shrinks as the step gets smaller.

Integrals: exact, numerical and applied

A definite integral accumulates a quantity over an interval: area under a curve, distance from velocity, work from force. Riemann sums show the idea directly by adding thin rectangles, and the Riemann Sum Calculator lets you watch the approximation improve as the rectangles narrow. The Numerical Integration Calculator uses better rules, such as Simpson's rule and Gaussian quadrature, that reach high accuracy with far fewer points.

Applied integrals have their own tools. Use Area Between Curves when two functions bound a region, Volume of Revolution for solids made by spinning a curve, and Arc Length for the length of a curve. The Improper Integral Calculator checks whether an integral over an infinite range or across a vertical asymptote converges at all.

Series and differential equations

A Taylor series replaces a function with a polynomial that matches it near one point, and a Fourier series represents a periodic function as a sum of sines and cosines. Before using any infinite series, the Series Convergence Calculator can test whether it converges and which test proves it.

Differential equations describe how a quantity changes in terms of itself, from cooling coffee to oscillating springs. The Differential Equation Solver handles standard analytic forms, while the ODE Numerical Solver steps through a solution with Euler, Heun or RK4. The Laplace Transform Calculator converts linear ODEs into algebra, which is the standard approach in control and circuit analysis.

Questions people ask

A symbolic answer is an exact formula, such as the derivative of x³ being 3x². A numerical answer is a decimal approximation at specific values, produced by a method with a known error. Numerical methods are essential when no closed-form answer exists.