Calculus
Derivatives, integrals, limits and series, handled symbolically where possible and numerically otherwise. 41 tools, including Antilog Calculator (inverse logarithm), Arc length calculator, Average rate of change calculator and Bernoulli ODE Solver. All free, with no account and nothing to install.
Antilog Calculator (inverse logarithm)The antilogarithm undoes the logarithm: antilog_b(y) = bʸ, so log_b(bʸ) = y. Pick base 10, e, 2 or your own, enter the exponent, and the result comes back with the inverse relationship spelled out and the round trip checked. Bases of zero or less, and base 1, are rejected with the reason.Arc length calculatorThe length of a circular arc from the radius and the central angle — arc = r·θ, with θ in radians. Work in either direction: find the arc from radius and angle, the radius from arc and angle, or the angle from arc and radius. It also returns the sector area and the chord length.Average rate of change calculatorCompute the average rate of change of a function on an interval: [f(b) − f(a)] / (b − a), the slope of the secant line between (a, f(a)) and (b, f(b)). Type any function of x, set the endpoints a and b, and see f(a), f(b) and the average rate instantly.Bernoulli ODE SolverSolves y′ + P(x)y = Q(x)yⁿ by showing the substitution v = y^(1−n) that turns it into a linear equation, then integrating. The two degenerate cases are handled explicitly: n = 0 is already linear, n = 1 is separable.Convolution CalculatorLinear and circular discrete convolution with the sliding-and-summing steps shown term by term, plus a symbolic continuous convolution solved exactly through the Laplace convolution theorem.Curl calculatorCompute the curl of a vector field: the scalar curl ∂Q/∂x − ∂P/∂y in 2D, or the full vector curl ∇×F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩ in 3D. Enter each component in x, y, z and get every partial symbolically plus the curl evaluated at an optional point.Curvature calculatorFor a curve y = f(x), the curvature κ = |f″| / (1 + f′²)^(3/2) measures how sharply it bends at a point, and the radius of curvature is its reciprocal. Type a function and a point; the derivatives are taken symbolically, then evaluated.Derivative CalculatorCompute single, partial, implicit and directional derivatives symbolically.Direction Field / Slope Field PlotterDraws f(x, y) as a grid of tangent dashes, so the shape of every solution is visible before a single one is computed. Add initial conditions and each spawns an RK4 curve that stays tangent to the field.Directional Derivative CalculatorCompute the directional derivative Dûf = ∇f · û at a point along a direction.Divergence calculatorCompute the divergence ∇·F = ∂P/∂x + ∂Q/∂y (+ ∂R/∂z) of a 2D or 3D vector field. Type each component as an expression in x, y, z — the tool shows every partial derivative symbolically, the divergence in closed form, and its numeric value at an optional evaluation point.Double Integral CalculatorNumerically evaluate a double integral ∬ f(x,y) over a rectangular region.Euler's Method CalculatorSteps through Euler's method one row at a time and — the whole point of the method — shows how far it drifts from the exact solution. Give the exact y(x) and the error column appears; halve the step and watch the error halve with it.Exponential Integral CalculatorCompute the exponential integral Ei(x) for any real x ≠ 0 to a chosen precision, using the convergent series γ + ln|x| + Σ xᵏ/(k·k!) or the asymptotic series for large arguments.Fast Fourier Transform (FFT) CalculatorRadix-2 Cooley–Tukey FFT of real or complex samples (3+4i, −2i): magnitude, phase, frequency bins and a bar plot, with Hann, Hamming and Blackman windows corrected for coherent gain and one-sided bins correctly doubled except DC and Nyquist.First-Order ODE SolverSolves dy/dx = f(x, y) from an initial condition with RK4, integrating outward in both directions, and plots the solution curve. Growth, decay, Newton cooling, logistic and separable cases are one click away.Fourier Series Coefficients Calculatora₀, aₙ and bₙ over any period, computed exactly by integration by parts when f is a polynomial and by Gauss–Legendre quadrature otherwise, with the truncated series plotted against f so the Gibbs overshoot is visible.Gradient calculator (multivariable)Compute the gradient ∇f = ⟨∂f/∂x, ∂f/∂y, …⟩ of a scalar function of several variables. List your variables, and the tool returns each partial derivative symbolically, the gradient vector in closed form, and ∇f evaluated at an optional point — the direction of steepest ascent.Implicit Derivative CalculatorDifferentiate an equation F(x,y) = 0 implicitly via dy/dx = −Fx/Fy (higher orders too).Improper Integral CalculatorEvaluate improper integrals with infinite bounds or discontinuities, with a convergence verdict.Instantaneous rate of change calculatorCompute the instantaneous rate of change f′(x₀) — the derivative of a function evaluated at a single point, and the slope of the tangent line there. Type any function of x and a point x₀; the tool differentiates symbolically, evaluates f′(x₀), and gives the tangent-line equation.Inverse Laplace Transform Calculatorf(t) from a rational F(s) by partial fractions, handling repeated roots and irreducible quadratics — 3/((s+2)²+9) comes back as a damped sine — with every residue listed and the expansion checked against F(s) numerically.Laplace Transform CalculatorF(s) from f(t) using the standard table plus first shifting, second shifting, multiplication by a polynomial in t and product-to-sum — and an explicit refusal, with the reason, for anything the table does not cover.Line Integral CalculatorCompute scalar (∫ f ds) and vector (∫ F·dr) line integrals along a parametric curve in 2D or 3D.Maclaurin Series CalculatorThe Taylor series at 0 for twelve standard functions — sin, cos, eˣ, ln(1+x), √(1+x), arcsin, arctan and more — with exact fractions such as 1/120, the general term and the error at a chosen x.Newton's method calculatorFind a root of f(x) = 0 with Newton's method: x_{n+1} = x_n − f(x_n) / f′(x_n). Enter a function, a starting guess, a tolerance and a maximum number of iterations; the tool differentiates f symbolically and shows the full iteration table converging to the root.Numerical Integration CalculatorCompare Gauss-Legendre, Romberg and adaptive Simpson on the same integral: each value, its function evaluations and its estimated error.Partial Derivative CalculatorCompute partial and mixed derivatives of a multivariable function symbolically.Partial Fraction Decomposition CalculatorBreaks a rational function into simple fractions, covering distinct linear factors, repeated factors and irreducible quadratics. The setup and the linear system solved for the coefficients are both shown, since that is the part being learned.Radius of Convergence CalculatorEnter aₙ as a formula in n or as a list of coefficients and get R by the ratio or root test — plus the endpoint behaviour most calculators skip: Σ(−1)ⁿ/(n+1) converges at x = 1 and diverges at x = −1.Related rates solverSolve classic related-rates problems from a template: expanding sphere, spreading ripple, inflating balloon, sliding ladder, filling cone, shadow from a lamppost, two vehicles separating, and a changing rectangle. Pick a scenario, enter the known quantities and rates, and read off the unknown rate with its governing equation.Riemann Sum CalculatorApproximate a definite integral by left, right, midpoint, trapezoidal or Simpson sums, drawn against the curve with a per-interval table.Runge-Kutta (RK4) Method CalculatorShows the four stage slopes k₁ to k₄ at every step and how the 1-2-2-1 weighting cancels the error through h⁴. Halve the step and the error drops about sixteenfold — the measured order is printed so you can watch it approach 4.Second-Order ODE SolverSolves a·y″ + b·y′ + c·y = g(x) with two initial conditions. Gives the characteristic roots, names the regime (undamped, under-damped, critically damped, over-damped), derives a particular solution for constant, polynomial and exponential forcing, and plots the result.Simpson's Rule CalculatorIntegrate with the 1/3 and 3/8 rules, with the weight pattern shown, the subinterval count adjusted when it must be, and the error bound.Single Variable Derivative CalculatorDifferentiate f(x) symbolically to any order, with optional evaluation at a point.Surface Integral CalculatorCompute scalar (∬ f dS) and flux (∬ F·dS) surface integrals over a parametrized surface.Taylor Series CalculatorType any f(x) — tan(x), sqrt(1+x), ln(1+x), exp(x)/cos(x) — pick the expansion point and get exact rational coefficients up to order 20, the derivatives f⁽ⁿ⁾(a) and the detected general term.Triple Integral CalculatorNumerically evaluate a triple integral ∭ f(x,y,z) over a rectangular box.Wronskian calculatorThe Wronskian of a set of functions — the determinant built from them and their successive derivatives — is the classic test for linear independence in differential equations. Enter your functions and a point; it differentiates each symbolically, builds the matrix and evaluates the determinant.Z-Transform Calculator (with ROC)Both directions: a sequence such as 0.5^n, n·0.8^n, cos(πn/4) or a finite sample list into X(z) with poles, zeros and the causal ROC, and a rational X(z) — q accepted as shorthand for z⁻¹ — back into causal samples and a closed form.