Calculus Solver
Differentiate, integrate, and evaluate limits — type an equation with a true math editor, see every rule applied step by step, and plot the function alongside its result.
What this tool covers
Calculus has three core operations this solver handles: differentiation (finding rates of change), integration (finding accumulated totals or reversing differentiation), and limits (describing behavior as a variable approaches a value). All three are connected by the Fundamental Theorem of Calculus, which says differentiation and integration are inverse operations of each other.
Differentiation rules
A derivative measures the instantaneous rate of change of a function. Every derivative in this tool is built from a small set of rules, applied recursively from the outermost structure of the expression inward:
Power Rule
\(\frac{d}{dx}[x^n] = n x^{n-1}\)
Sum / Difference Rule
\(\frac{d}{dx}[f \pm g] = f' \pm g'\)
Product Rule
\(\frac{d}{dx}[fg] = f'g + fg'\)
Quotient Rule
\(\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}\)
Chain Rule
\(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
Exponential Rule
\(\frac{d}{dx}[e^x] = e^x\)
Natural Log Rule
\(\frac{d}{dx}[\ln x] = \frac{1}{x}\)
Logarithmic Differentiation
For \(f(x)^{g(x)}\), take \(\ln\) of both sides first
Integration techniques
Integration is harder than differentiation because there's no single mechanical procedure that always works — instead, this tool tries a sequence of recognized patterns, in order of simplicity, until one applies:
| Technique | When it applies |
|---|---|
| Power Rule | \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\), for \(n \neq -1\) |
| Basic antiderivatives | Direct lookup for \(\sin x\), \(\cos x\), \(e^x\), \(\frac{1}{x}\), \(\sec^2 x\), \(\frac{1}{1+x^2}\), and similar standard forms |
| Sum / Constant Multiple Rules | Splits sums into separate integrals; pulls constant factors outside the integral sign |
| u-Substitution | Detects when one factor is the derivative of an inner function elsewhere in the expression, e.g. \(2x \cdot \sin(x^2)\) |
| Integration by Parts | For products of a polynomial and \(\sin x\), \(\cos x\), or \(e^x\) — applies \(\int u \, dv = uv - \int v \, du\), recursing if needed |
Limits and indeterminate forms
A limit describes what value a function approaches as the input gets arbitrarily close to some point — without necessarily ever reaching it. This matters most when direct substitution gives an undefined result, like 0/0.
L'Hôpital's Rule
When a limit produces the indeterminate form 0/0 or ∞/∞, L'Hôpital's Rule says you can differentiate the numerator and denominator separately and try the limit again:
This solver applies L'Hôpital's Rule automatically when it detects an indeterminate form, and will keep reapplying it if the result is still indeterminate.
One-sided limits and limits at infinity
This tool checks the limit from both the left and right side of the approached value. If they disagree, the two-sided limit does not exist — a subtlety that's easy to miss when only checking one direction. Limits as x → ∞ or x → −∞ describe long-run behavior, useful for finding horizontal asymptotes.
Using the solver
- Choose an operationPick Derivative, Integral, or Limit from the sidebar. Each mode reveals the fields it needs.
- Type your function using the math editorClick into the equation field and type naturally — typing x^2 automatically becomes a proper exponent, sqrt opens a radical, and a small on-screen keyboard icon gives you one-tap access to fractions, Greek letters, and trig functions if you'd rather not type LaTeX-style shortcuts.
- Set the variableMost problems use x, but you can change this if your function uses a different variable name.
- For limits, set the approach valueType the value x is approaching — including ∞ or -∞ for limits at infinity, available from the on-screen keyboard's symbols panel.
- Press SolveThe result appears immediately, with every rule used shown underneath in order, and a graph of the original function plotted alongside the result where applicable.
- Try an example chipIf you're not sure what to type, click any example below the equation field to load a ready-made problem for the current mode.
Reading the step-by-step trace
Each step in the trace shows three things: the rule name that was applied (e.g. "Product Rule" or "u-Substitution"), a short explanation of the rule in symbolic form, and the sub-expression it was applied to. Steps are listed in the order the engine actually processed them, which is generally outside-in — so for a nested expression like sin(x²), you'll see the Chain Rule step before the inner Power Rule step that handles x² itself.
Supported syntax reference
You don't need to learn special codes — the equation field formats things as you type. The table below shows exactly what to type on a physical keyboard and what it turns into. You can also tap the small keyboard icon inside the equation field for buttons that insert these directly.
| What it is | Type this | Renders as |
|---|---|---|
| Power / exponent | x^2 | \(x^2\) |
| Fractional exponent | x^(1/2) | \(x^{1/2}\) |
| Fraction | x/y or / after a term | \(\frac{x}{y}\) |
| Square root | sqrt then type the contents | \(\sqrt{x}\) |
| Multiplication | x*y or just xy | \(x \cdot y\) |
| Sine, cosine, tangent | sin(x), cos(x), tan(x) | \(\sin(x)\), \(\cos(x)\), \(\tan(x)\) |
| Secant, cosecant, cotangent | sec(x), csc(x), cot(x) | \(\sec(x)\), \(\csc(x)\), \(\cot(x)\) |
| Inverse trig | asin(x), acos(x), atan(x) | \(\arcsin(x)\), \(\arccos(x)\), \(\arctan(x)\) |
| Hyperbolic | sinh(x), cosh(x), tanh(x) | \(\sinh(x)\), \(\cosh(x)\), \(\tanh(x)\) |
| Natural log | ln(x) | \(\ln(x)\) |
| Base-10 log | log(x) | \(\log(x)\) |
| Exponential | e^x | \(e^x\) |
| Pi | pi, or the π key on the on-screen keyboard | \(\pi\) |
| Euler's number | e (on its own, not part of a word) | \(e\) |
| Infinity (for limits) | the ∞ key on the on-screen keyboard | \(\infty\) |
| Grouping | ( and ) | \(\left(x+1\right)\) |
Using the on-screen keyboard
Tap the small keyboard icon inside the equation field to bring up an on-screen math keyboard — useful for symbols with no obvious keyboard key, like fractions, roots, π, and infinity. Tap the icon again, or tap anywhere outside the field, to dismiss it. You can also type directly with your physical keyboard at any time, using the keystrokes shown in the table above.
About the Solve button
Loading an example (from a chip or the dropdown) solves it immediately, so you see a result right away. The Solve button is for whenever you've typed or edited your own expression — press it any time after changing the function, the variable, or the approach value to get an updated result.
Suggested classroom uses
- Verify homework by hand, then check. Have students work a derivative or integral on paper first, then check their answer and compare their working against the rule trace.
- Predict the rule before solving. Show the unsolved expression and ask students to name which differentiation or integration rule applies before revealing the trace.
- Explore L'Hôpital's Rule visually. Try a limit like sin(x)/x as x→0, watch the solver detect the 0/0 form automatically, and discuss why direct substitution fails there.
- Discuss when integration fails. Try an expression with no elementary antiderivative and use the "no closed form found" result as a launching point for discussing series-based or numerical approaches.