All Calculus Calculators
Every Calculus Solver tool, organized by topic. We're steadily growing this library toward 100+ calculators, jump to a category below or use search in the header.
Derivatives
Differentiate functions step by step, from basic power-rule problems to implicit and higher-order derivatives.
Derivative Calculator
Differentiate any function and see every step, from the power rule to the chain rule.
Second Derivative Calculator
Find f''(x) instantly and see how it reveals concavity and inflection behavior.
Implicit Differentiation Calculator
Find dy/dx for equations where y isn't isolated, like circles and other implicit curves.
Chain Rule Calculator
Differentiate composite functions like sin(3x²) with the outer and inner parts worked separately.
Product Rule Calculator
Differentiate products like x²·sin(x) with u, v, u', and v' worked out separately.
Quotient Rule Calculator
Differentiate fractions like sin(x)/x² with the numerator and denominator handled separately.
Logarithmic Differentiation Calculator
Differentiate variable-to-variable powers like x^x by taking logarithms first.
Higher-Order Derivative Calculator
Find the nth derivative of a function with every intermediate derivative listed in order.
Derivative at a Point Calculator
Evaluate the instantaneous rate of change f'(a) at a specific x-value, verified numerically.
Parametric Derivative Calculator
Compute dy/dx = y'(t)/x'(t) for parametric curves, with vertical tangents flagged.
Polar Derivative Calculator
Find the Cartesian slope of a polar curve at any θ with the polar slope formula.
Implicit Second Derivative
From F(x, y) = 0 straight to d²y/dx², concavity of implicit curves without solving for y.
Average Rate of Change
The difference quotient over an interval, drawn as a secant and compared to instantaneous rates.
Secant Line Calculator
Slope and equation of the line through two points on a curve, the approach to the tangent.
Limit Definition of Derivative
The difference quotient tabulated toward its limit, confirming the derivative or a corner.
Differentiability Calculator
Compare left and right derivatives at a point to detect corners, cusps, and vertical tangents.
Difference Quotient Calculator
Form (f(x+h) − f(x))/h, shrink h step by step, and watch the secant slope become the derivative.
Inverse Function Derivative
Find (f⁻¹)'(b) from the reciprocal-slope rule, no inverse formula required.
Parametric Second Derivative
Find d²y/dx² for x(t), y(t) with every quotient shown, and read the curve's concavity.
Integrals
Find indefinite and definite integrals, approximate area under curves, and work through integration techniques.
Indefinite Integral Calculator
Find antiderivatives of common functions with the rule used at each step, plus the constant of integration.
Definite Integral Calculator
Evaluate a definite integral between two limits and see the shaded area under the curve.
Riemann Sum Calculator
Approximate area under a curve with left, right, or midpoint rectangles and compare to the exact integral.
Trapezoidal Rule Calculator
Approximate integrals with trapezoids and compare the result against the exact value.
Simpson's Rule Calculator
Approximate integrals with parabolic arcs, far more accurate than rectangles for the same n.
Improper Integral Calculator
Evaluate integrals to infinity by watching partial integrals converge, or detect divergence.
U-Substitution Calculator
Integrate composite patterns like 2x·cos(x²) with u and du identified automatically.
Integration by Parts Calculator
Integrate products like x·eˣ with u and dv chosen and every round of the formula shown.
Trig Integral Calculator
Odd-power substitution, power reduction, and product-to-sum, the right identity chosen for you.
Partial Fractions Calculator
Decompose p(x)/q(x) by the cover-up method and integrate every piece to a logarithm.
Romberg Integration Calculator
Accelerate trapezoid estimates with Richardson extrapolation and read the full Romberg triangle.
Gaussian Quadrature Calculator
Evaluate integrals with Gauss-Legendre nodes and weights, 2 to 5 points, error included.
Midpoint Rule Calculator
Sum midpoint-height rectangles, see them drawn, and compare accuracy against the trapezoidal rule.
Monte Carlo Integration
Integrate by random sampling and watch the 1/√N error law in action, sample by sample.
Integral of |f(x)|
Split at every sign change, flip the negative pieces, and integrate |f| the way the textbook intends.
Gamma Function Calculator
Evaluate Γ(x), connect it to factorials, and explore its poles and Stirling growth on the graph.
Beta Function Calculator
Evaluate B(a, b) by the gamma identity and verify it against the defining integral.
Error Function Calculator
Evaluate erf(x), erfc(x), and the normal-distribution probabilities they encode.
Limits & Continuity
Evaluate limits at a point, at infinity, and one-sided limits, and test functions for continuity.
Limit Calculator
Evaluate limits at a point, at infinity, or one-sided limits, including 0/0 indeterminate forms.
One-Sided Limit Calculator
Evaluate left-hand and right-hand limits separately and see whether they agree.
Limit at Infinity Calculator
Find end behavior as x grows without bound and identify horizontal asymptotes.
L'Hopital's Rule Calculator
Resolve 0/0 and ∞/∞ indeterminate forms with each differentiation round shown.
Continuity Calculator
Run the three-part continuity test at a point and classify any discontinuity found.
Squeeze Theorem Calculator
Verify the sandwich numerically, squeeze the bounds together, and watch oscillations get crushed.
Piecewise Limit Calculator
Left and right limits from their own pieces, with an exists-or-jump verdict at the seam.
Epsilon-Delta Calculator
Challenge with an ε, get a working δ, the formal definition of a limit as a playable game.
IVT Calculator
Verify the IVT hypotheses and locate a point c where f(c) hits any target value between f(a) and f(b).
Discontinuity Calculator
Scan a function for holes, jumps, and infinite discontinuities and classify each with one-sided limits.
Series & Sequences
Expand functions into Taylor and Maclaurin series and test sequences and series for convergence.
Taylor Series Calculator
Expand a function into a Taylor or Maclaurin polynomial and see how the approximation compares to the original.
Maclaurin Series Calculator
Expand a function around 0 with every coefficient's derivation shown.
Geometric Series Calculator
Sum a + ar + ar² + … with the convergence test and the a/(1−r) infinite sum.
Partial Sum Calculator
Sum any Σ a(n) numerically with running partial sums plotted term by term.
Ratio Test Calculator
Test series convergence with lim |a(n+1)/a(n)|, sampled and plotted against 1.
Root Test Calculator
Test convergence with lim |a(n)|^(1/n), the tool of choice for nth-power terms.
Alternating Series Calculator
Run the alternating series test and sum Σ(−1)ⁿ⁺¹bₙ with its error bound.
p-Series Calculator
Test Σ1/nᵖ with the p > 1 rule and watch partial sums settle or drift.
Sequence Convergence Calculator
Find lim a(n) as n → ∞ with sampled values and the first terms plotted.
Radius of Convergence Calculator
Find R for Σcₙxⁿ from the coefficient ratio limit, with the interval stated.
Taylor Remainder Calculator
Compute the Lagrange error bound for a Taylor polynomial and compare with the true error.
Binomial Series Calculator
Expand (1+x)^k for any real k, including √(1+x) and 1/(1+x), term by term.
Fourier Series Calculator
Coefficients by quadrature, symmetry detected, and the partial sum plotted against the true wave.
Interval of Convergence
Radius from the ratio test plus endpoint tests for the exact open or closed interval.
Integral Test Calculator
Check the integral test hypotheses, evaluate the improper integral, and decide convergence.
Limit Comparison Test Calculator
Compare aₙ against a benchmark bₙ, compute the limit of the ratio, and share the verdict.
Alternating Series Error Bound
Truncate an alternating series and bound the remainder with the first omitted term.
Telescoping Series Calculator
Watch interior terms cancel in pairs and read the exact sum from the two survivors.
Divergence (nth-Term) Test
Check lim aₙ first: a nonzero limit ends the story, a zero limit sends you to a sharper test.
Direct Comparison Test
Verify a termwise inequality against a benchmark p-series or geometric series and inherit its verdict.
Absolute vs Conditional
Sort a series into absolutely convergent, conditionally convergent, or divergent, with the tests shown.
Padé Approximant Calculator
Turn a Taylor series into a rational approximant that often outruns the polynomial it came from.
Multivariable Calculus
Partial derivatives, gradients, and other tools for functions of two or more variables.
Partial Derivative Calculator
Differentiate multivariable functions with respect to x or y while holding the other variable constant.
Double Integral Calculator
Integrate f(x, y) over a rectangle with the iterated setup and a precise numeric result.
Gradient Calculator
Compute ∇f = ⟨fx, fy⟩ symbolically and read off the steepest-ascent direction.
Directional Derivative Calculator
Find the rate of change in any direction via D_u f = ∇f · u, fully worked.
Second Partials Calculator
Compute fxx, fxy, fyx, fyy and watch Clairaut's mixed-partial equality hold.
Hessian Matrix Calculator
Build the 2×2 Hessian, get D = fxx·fyy − fxy², and classify bowl, dome, or saddle.
Jacobian Calculator
Form the Jacobian of (u, v) and compute det J, the local area scale factor.
Tangent Plane Calculator
Find the plane touching z = f(x,y) at a point, plus its normal vector.
Lagrange Multipliers
Solve ∇f = λ∇g on any constraint curve, every tangency point found, max and min declared.
Multivariable Critical Points
Solve ∇f = 0 and classify every point with the discriminant D, minima, maxima, saddles.
Triple Integral Calculator
Integrate f(x, y, z) over a box region with the full iterated setup and a high-accuracy numeric result.
Laplacian Calculator
Compute ∇²f = fxx + fyy symbolically, evaluate it at a point, and check if f is harmonic.
Multivariable Chain Rule Calculator
Find dz/dt for z = f(x, y) along a path (x(t), y(t)) with every partial and product shown.
Total Differential Calculator
Form dz = fx dx + fy dy, estimate the change in f, and compare it with the true change.
Implicit Partial Derivatives
Differentiate a surface F(x, y, z) = 0 implicitly and evaluate both slopes at any point.
Quadratic Approximation
Assemble the two-variable quadratic Taylor model from the gradient and Hessian at your point.
Surface Area (Double Integral)
Integrate √(1 + fx² + fy²) over the base rectangle to measure a curved surface's true area.
Level Curves Calculator
Slice a surface at constant heights and read its hills, valleys, and passes from the contour map.
Gradient Descent Calculator
Iterate x ← x − η∇f and watch the optimization path bend downhill toward a minimum.
Polar Double Integral
Integrate over disks, sectors, and annuli with the r dr dθ element handled for you.
Spherical Triple Integral
Integrate over balls, shells, and cones with the ρ² sin φ volume element built in.
Applications of Derivatives
Use derivatives to find critical points, extrema, concavity, tangent lines, and optimize real functions.
Critical Points Calculator
Locate critical points and classify each as a local max, local min, or neither.
Inflection Point Calculator
Find where a function changes concavity and view concave-up and concave-down intervals.
Tangent Line Calculator
Find the equation of the line tangent to a curve at any point, with the slope worked from the derivative.
Normal Line Calculator
Find the line perpendicular to a curve at a point using the negative reciprocal slope.
Linear Approximation Calculator
Approximate a function near a point with its linearization L(x) and see the estimation error.
Newton's Method Calculator
Find roots iteratively with tangent lines, watching each guess improve in a table.
Mean Value Theorem Calculator
Find where the tangent slope equals the secant slope over an interval, as MVT guarantees.
Position-Velocity-Acceleration Calculator
Turn s(t) into v(t) and a(t), then read off motion at any instant.
Related Rates Calculator
Differentiate any relation through time and solve for the unknown rate at the given instant.
Optimization Calculator
The closed-interval method end to end: critical points, candidate table, global max and min.
Curve Sketching Calculator
The whole checklist, critical points, monotonicity, inflections, concavity, end behavior.
Concavity Calculator
Concave up and down intervals plus verified inflection points from the second derivative.
Curvature Calculator
Curvature and radius of curvature at a point, with the osculating circle drawn to scale.
Bisection Method Calculator
Bracket a root and halve the interval step by step, with a full iteration table and error bound.
Secant Method Calculator
Find roots with secant lines through two guesses, no derivative needed, iterations tabulated.
Fixed-Point Iteration Calculator
Iterate x = g(x), test convergence with |g'(x*)| < 1, and watch the cobweb diagram.
Rolle's Theorem Calculator
Confirm f(a) = f(b) and find every interior point where the tangent goes horizontal.
Absolute Extrema Calculator
Run the closed-interval method: compare critical points and endpoints to crown the global max and min.
Increasing/Decreasing Intervals
Split the line at critical points and report exactly where f rises and where it falls.
First Derivative Test
Check the sign of f' on each side of every critical point and classify it as max, min, or neither.
Second Derivative Test
Evaluate f'' at each critical point to sort local maxima from minima in one step.
Applications of Integrals
Use integrals to compute arc length, area, volume, work, and averages.
Arc Length Calculator
Find the length of a curve between two points using the arc length integral formula.
Average Value Calculator
Find a function's mean height over an interval with the integral average-value formula.
Area Between Curves Calculator
Compute the area enclosed between two curves with the region shaded on a live graph.
Disk Method Calculator
Rotate a curve around the x-axis and find the solid's volume with V = π∫f² dx.
Washer Method Calculator
Rotate the region between two curves and find the hollow solid's volume with washers.
Shell Method Calculator
Rotate around the y-axis and find volume with cylindrical shells: V = 2π∫x·f(x) dx.
Surface of Revolution Calculator
Find the area of the surface swept out when a curve spins about the x-axis.
Work (Variable Force) Calculator
Integrate a position-dependent force to find total work, e.g. stretching springs.
Center of Mass Calculator
Find where a rod with varying density balances, from mass and moment integrals.
Total Distance Calculator
Integrate |v(t)| properly, split at sign changes, and compare with net displacement.
Polar Area Calculator
Sweep a polar curve r(θ) through an angle range and compute the enclosed area.
Polar Arc Length Calculator
Measure the length of a polar curve with √(r² + (dr/dθ)²) integrated over θ.
Parametric Arc Length Calculator
Integrate speed √(x'² + y'²) along a parametric path to find its true length.
Volume by Cross Sections
Stack square, semicircular, or triangular slices on a base region and integrate the volume.
Area Under Curve
Get both answers: the signed integral and the true geometric area with below-axis lobes counted positive.
Hydrostatic Force Calculator
Integrate pressure times strip width over a submerged plate to get the total fluid force.
Moment of Inertia (Calculus)
Integrate r² dm for rods with variable density or plane regions, radius of gyration included.
Centroid Calculator
Locate (x̄, ȳ) of a region between curves by moment integrals, drawn right on the region.
Gini Coefficient Calculator
Measure income inequality: integrate a Lorenz curve and read the Gini index off the shaded gap.
Consumer & Producer Surplus
Find market equilibrium, then integrate the demand and supply gaps into consumer and producer surplus.
PDF Calculator
Check that a density integrates to 1, then extract probabilities, mean, and variance from it.
Catenary Calculator
Compute sag and exact arc length of a hanging cable, and see how it differs from a parabola.
Differential Equations
Solve and visualize ordinary differential equations.
Initial Value Problem Calculator
Solve y' = f(x) with y(x₀) = y₀, find the antiderivative and pin down the constant C.
Euler's Method Calculator
Approximate y' = f(x, y) step by step, with every iteration listed and the error made visible.
Runge-Kutta Calculator
Fourth-order accuracy: k₁ through k₄ shown for every step of the RK4 iteration.
Slope Field Generator
See every solution at once: direction-field ticks with a traced solution curve through your point.
Separable ODE Calculator
Separate y' = f(x)·g(y), integrate both sides symbolically, and pin down C from an initial condition.
Exponential Growth & Decay
From dy/dt = ky to y₀e^(kt): half-life, doubling time, and the value at any moment.
Second-Order ODE Calculator
Characteristic roots, all three damping cases, and constants fitted to your initial conditions.
Logistic Growth Calculator
Solve dy/dt = ky(1 - y/M) into an S-curve, with the peak-growth inflection located.
Laplace Transform Calculator
Transform f(t) into F(s) term by term using the standard Laplace table, with every rule shown.
Wronskian Calculator
Build the Wronskian determinant of two or three functions and test their linear independence.
Linear First-Order ODE Calculator
Solve y' + P(x)y = Q(x) with the integrating factor method and graph the solution curve.
Exact Differential Equation Calculator
Check exactness of M dx + N dy = 0, build the potential F(x, y), and draw the solution curve.
Convolution Calculator
Slide, multiply, and integrate: compute f ∗ g on a grid and see the smoothing effect graphed.
Bernoulli Equation Calculator
Linearize yⁿ nonlinearity with the v = y^(1−n) substitution and solve the resulting equation.
Homogeneous First-Order ODE
Reduce dy/dx = F(y/x) to a separable equation with v = y/x and draw the solution family.
Phase Line Calculator
Find every equilibrium of y' = f(y) and classify its stability from the sign chart.
Linear System Phase Portrait
Compute eigenvalues, classify the equilibrium, and watch trajectories swirl through the phase portrait.
Predator-Prey Simulator
Simulate predator and prey populations cycling around their equilibrium, in time and phase space.
Improved Euler (Heun) Method
Predict with Euler, correct with the averaged slope, and watch the accuracy jump an order.
Picard Iteration Calculator
Build successive approximations by repeated integration and watch them converge to the solution.
Inverse Laplace Transform
Read F(s) against the Laplace table term by term and reassemble the time-domain signal f(t).
Newton's Law of Cooling
Fit k from your data, then predict temperatures and waiting times along the exponential approach.
Mixing (Tank) Problems
Set up and solve the tank equation from flow rates and concentrations, steady state included.
Vector Calculus
Gradients, divergence, curl, and line integrals for vector fields.
Divergence Calculator
Compute div F = Px + Qy and see sources and sinks on a live arrow plot.
Curl Calculator
Compute curl F = Qx − Py and read the field's local rotation off the arrows.
Line Integral Calculator
Integrate F · dr along a parametric path, the work done by a field along a curve.
Vector Field Plotter
Draw any field ⟨P, Q⟩ as arrows, with div and curl computed alongside.
Green's Theorem Calculator
Compute a circulation integral two ways, boundary line integral and double integral, and watch them match.
Conservative Field Calculator
Check Py = Qx, decide if the field is conservative, and recover its potential function.
Unit Tangent Vector Calculator
Find T(t) and the principal normal N(t) for a parametric curve, drawn right on the path.
Flux Integral Calculator
Measure outward flow across a closed curve and verify the divergence theorem numerically.
Scalar Line Integral
Integrate a scalar field along a curve with the ds element handled step by step.
Precalculus Prep
Foundational function analysis tools that support calculus coursework.
Domain & Range Calculator
Every restriction found and solved, denominators, roots, and logs, with the graph to confirm.
Composite Function Calculator
Both orders composed symbolically, f(g(x)) and g(f(x)), with two-stage numeric evaluation.
Inverse Function Calculator
Solve y = f(x) for x step by step, check one-to-one, and see the reflection across y = x.
Asymptote Calculator
Vertical, horizontal, and slant asymptotes found and drawn dashed under the curve.
Piecewise Function Grapher
Up to three pieces graphed without false bridges, with a continuity verdict at every seam.