Calculus Calculators That Show Every Step

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.

Partial Derivative Calculator

Differentiate multivariable functions with respect to x or y while holding the other variable constant.

Implicit Differentiation Calculator

Find dy/dx for equations where y isn't isolated, like circles and other implicit curves.

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.

Limit Calculator

Evaluate limits at a point, at infinity, or one-sided limits, including 0/0 indeterminate forms.

Taylor Series Calculator

Expand a function into a Taylor or Maclaurin polynomial and see how the approximation compares to the original.

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.

Arc Length Calculator

Find the length of a curve between two points using the arc length integral formula.

Riemann Sum Calculator

Approximate area under a curve with left, right, or midpoint rectangles and compare to the exact integral.

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.

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.

Derivative at a Point Calculator

Evaluate the instantaneous rate of change f'(a) at a specific x-value, verified numerically.

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.

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.

Average Value Calculator

Find a function's mean height over an interval with the integral average-value formula.

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.

Area Between Curves Calculator

Compute the area enclosed between two curves with the region shaded on a live graph.

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.

Double Integral Calculator

Integrate f(x, y) over a rectangle with the iterated setup and a precise numeric result.

Initial Value Problem Calculator

Solve y' = f(x) with y(x₀) = y₀, find the antiderivative and pin down the constant C.

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.

Position-Velocity-Acceleration Calculator

Turn s(t) into v(t) and a(t), then read off motion at any instant.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Implicit Second Derivative

From F(x, y) = 0 straight to d²y/dx², concavity of implicit curves without solving for y.

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.

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.

Fourier Series Calculator

Coefficients by quadrature, symmetry detected, and the partial sum plotted against the true wave.

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.

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.

Interval of Convergence

Radius from the ratio test plus endpoint tests for the exact open or closed interval.

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.

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.

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.

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.

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.

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.

Volume by Cross Sections

Stack square, semicircular, or triangular slices on a base region and integrate the volume.

IVT Calculator

Verify the IVT hypotheses and locate a point c where f(c) hits any target value between f(a) and f(b).

Rolle's Theorem Calculator

Confirm f(a) = f(b) and find every interior point where the tangent goes horizontal.

Discontinuity Calculator

Scan a function for holes, jumps, and infinite discontinuities and classify each with one-sided limits.

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.

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.

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.

Area Under Curve

Get both answers: the signed integral and the true geometric area with below-axis lobes counted positive.

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.

Convolution Calculator

Slide, multiply, and integrate: compute f ∗ g on a grid and see the smoothing effect graphed.

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.

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.

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.

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.

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.