Calculus Workbench
One calculus tool for everything: derivatives, integrals, limits, Taylor/Maclaurin series, critical & inflection points, area under a curve, arc length, related rates and optimization — with a live graph and step-by-step output. Exact symbolic math in your browser.
How to Use
- Type a function of one variable — use <code>^</code> for powers and the usual functions <code>sin, cos, tan, exp, ln, sqrt, asin, …</code> plus <code>pi</code> and <code>e</code>. Implicit multiplication works (<code>2x</code>, <code>x sin(x)</code>).
- Pick an operation: <strong>Derivative, Integral, Limit, Taylor series, Critical points, Inflection points, Area under curve, Arc length, Related rates,</strong> or <strong>Optimization</strong>.
- Extra fields appear when needed — bounds <code>a, b</code> for area/arc length/optimization, a centre for the Taylor series, an order for higher derivatives, a point and side for limits.
- Differentiation is fully symbolic and exact; integration handles polynomials, standard forms, linear substitution and integration by parts, and falls back to a numeric value for the definite case.
- The graph plots your function (with critical/inflection points marked and the area shaded), and the <strong>Steps</strong> panel shows the working. For <strong>Related rates</strong>, enter an equation with <code>=</code> (e.g. <code>A = pi*r^2</code>) — it differentiates both sides with respect to time.
What each tool does
About the Calculus Workbench
Calculus Workbench is a quick, free tool for everyday maths and number work. It works in your browser and keeps everything on your device. One calculus tool for everything: derivatives, integrals, limits, Taylor/Maclaurin series, critical & inflection points, area under a curve, arc length, related rates and optimization — with a live graph and step-by-step output. Exact symbolic math in your browser.
How it works
Enter your figures and the result appears instantly, updating the moment you change anything. There is no submit button and nothing to wait for, so it is easy to try a few what-if numbers and compare the results. Just check each box holds the kind of value it expects.
Want the deeper story? The Knowledge Base explains the ideas behind the tools in more detail.
Frequently Asked Questions
What can the derivative do?
Full symbolic differentiation with the sum, product, quotient and chain rules over polynomials, rational functions, roots, exponentials, logarithms and all the trig / inverse-trig / hyperbolic functions. Set an order to get the 2nd, 3rd, … derivative. The result is exact — e.g. <code>x³ − 6x² + 9x</code> gives <code>3x² − 12x + 9</code>.
How good is the integral?
It computes exact antiderivatives for polynomials (term by term), the standard table (sin, cos, eˣ, 1/x → ln, 1/(1+x²) → arctan, …), linear substitution <code>f(ax+b)</code>, and integration by parts for polynomial × {eˣ, sin, cos} and <code>ln x</code>. When no elementary antiderivative is found it says so — use <strong>Area under curve</strong> for a guaranteed numeric value (Simpson's rule).
How are limits evaluated?
Numerically but carefully: the function is sampled from the left and right approaching the point at successively smaller distances, so removable discontinuities like <code>(x²−1)/(x−1) → 2</code> and <code>sin(x)/x → 1</code> are found, one-sided limits are supported, and limits at ±∞ are estimated from large values. If the two sides disagree it reports that the limit does not exist.
What does "critical points" report?
It differentiates, solves f′(x) = 0 (exactly when f′ is a polynomial, numerically otherwise), and classifies each point with the second-derivative test as a local maximum, local minimum, or a flat/inflection candidate, giving the (x, y) coordinates. Inflection points solve f″(x) = 0 and keep only where concavity actually changes sign.
How does the optimization tool work?
Enter the objective function and, optionally, an interval [a, b]. It finds the interior critical points, classifies them, and — if you give an interval — also evaluates the endpoints, then reports the global maximum and minimum over the domain. It's the standard closed-interval method for optimization word problems.
Is anything sent to a server?
No. The whole engine — parser, symbolic differentiation, integration, the numeric routines and the graph — runs locally in your browser. It works offline and nothing is uploaded.
How do I use the Calculus Workbench?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Calculus students
Check derivatives and integrals, see the steps, and visualise the function and its key points.
Physics & engineering
Related rates, optimization, and arc length for real problems; differentiate and integrate models exactly.
Exam prep
Confirm critical points, inflection points, and Taylor expansions quickly with a graph to back them up.
Series & approximation
Build Taylor/Maclaurin series to any order and compare against the curve.
Numerical work
Get definite integrals (area) and arc lengths even when no closed form exists.
Teaching
Demonstrate the second-derivative test, concavity, and the closed-interval optimization method live.
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