Derivative & Integral Calculator with Steps (Limits & Taylor Series)

See full step-by-step derivatives, integrals, and limits rendered in proper math notation.

The solver leans on nerdamer, a JavaScript symbolic algebra engine, for derivatives, integrals, limits and a hand-built Taylor expansion — all computed synchronously in your browser with no server round trip. Because nerdamer only ever hands back a final simplified expression, this tool layers its own step-by-step narrative on top, classifying which textbook rule applies and showing the intermediate transformation before the final answer.

How to use the derivative and integral calculator

Type the function in the f(x) field, for example sin(x^2), and change the Variable box if your function uses t, u or another letter.
Choose Derivative, Integral, Limit or Taylor series, or click one of the example chips to load a ready-made problem.
Set the options for that operation: 1st or 2nd derivative; an indefinite (+C) or definite a-to-b integral; a limit point with two-sided, left-hand or right-hand direction; or an expansion point and a Taylor order from 0 to 12.
Read the numbered steps, each labelled with the rule or action applied, followed by the typeset result; the answer updates as you edit, with no Calculate button to press.

Derivatives, integrals and named rules

Before running nerdamer's differentiation, the input expression is classified against the rules you'd recognize from a calculus course — sum, product, quotient, chain, power — by inspecting its top-level structure, so the step panel can say which rule is in play rather than just showing symbols moving around. First and second derivatives are both supported. Integration follows the same pattern: bounded definite integrals and unbounded indefinite integrals both get a rule label and nerdamer's simplified antiderivative.

Not every integral nerdamer can technically compute gets returned as a usable answer. If the result contains a special function with no elementary closed form — the error function erf, the sine and cosine integrals Si and Ci, the exponential integral Ei, the Fresnel integrals, the dilogarithm, the Lambert W function, the Riemann zeta function, and a handful of others — the tool reports the integral as unsupported by name instead of handing back a string most students have never seen and can't use.

Derivative calculator input syntax

Use ^ for powers and write multiplication with *: x^3 - 2*x^2 + 5, or x^2*exp(x).
Put function arguments in parentheses: sin(x^2), cos(3*x), sqrt(x+1), exp(x) for e to the x, and log(x) for the natural logarithm.
Group numerators and denominators explicitly, such as (x^2 - 1)/(x - 1), so the quotient is read the way you intend.
For a limit at infinity type Infinity (or -Infinity) as the limit point; definite-integral bounds and the Taylor expansion point take plain numbers.
Any single-letter or longer variable name works as long as the same name is typed into the Variable field.

Limits, including one-sided behavior nerdamer can't give you directly

nerdamer's own limit computation is two-sided only. For a limit that resolves to a 0/0 or similarly indeterminate form at the target point, this tool applies L'Hopital's rule directly — differentiating numerator and denominator and re-evaluating — since that result is the same from either direction. For a limit that blows up to a nonzero value over zero, direction actually matters, so the solver numerically probes the function from the left and from the right using a shrinking sequence of offsets around the target point and reports each side's behavior — approaching positive infinity, negative infinity, a finite value, or undefined — separately rather than papering over a genuine two-sided disagreement.

Taylor series, built by hand

nerdamer has no built-in Taylor expansion function, so this solver constructs one from the textbook definition directly: it repeatedly differentiates the expression up to the requested order, evaluates each derivative at the expansion point, and assembles the polynomial term by term as f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! and so on. The order is capped at 12 in the interface — high enough to see a series converge visibly around a well-behaved point, low enough that the repeated symbolic differentiation stays fast.

Where symbolic math like this runs out of road

An antiderivative that genuinely has no elementary closed form is reported as unsupported rather than returned as an unfamiliar special function — useful for knowing when to reach for numerical integration or a table of known transforms instead.
The step narrative names which differentiation or integration rule applies from the expression's surface structure, so an unusual or deliberately obscure formatting of an otherwise ordinary expression can occasionally get labeled with the wrong rule name even though the underlying computed result is still correct.
A Taylor series is only a local approximation — it can diverge badly away from the expansion point for functions with a limited radius of convergence, and the tool does not warn you where that radius ends.

Worked calculus examples with rule labels

d/dx of x^2*sin(x): labelled Product rule, with the result 2x·sin(x) + x²·cos(x).
d/dx of sin(x^2): labelled Chain rule (outer derivative times inner derivative), giving 2x·cos(x²).
∫ x^2*exp(x) dx: the Method step reads 'Product of functions — computer algebra system applies substitution/by-parts', and the antiderivative is equivalent to eˣ(x² − 2x + 2) + C.
lim sin(x)/x as x → 0: direct substitution shows 0/0, L'Hopital's rule rewrites it as cos(x)/1, and the result is 1.
lim 1/x as x → 0: two-sided reports that the limit does not exist because the left side heads to −∞ and the right to +∞; choosing left-hand or right-hand returns the matching infinity.
Taylor series of exp(x) about 0 at order 4: 1 + x + x²/2 + x³/6 + x⁴/24.

Questions people ask about Step-by-Step Symbolic Calculus Solver

Why does the tool say my integral is unsupported instead of giving an answer?: Certain integrals have no closed form built from ordinary elementary functions — they only resolve to named special functions like the error function or the sine integral. Rather than hand back an unfamiliar symbol most readers can't use, the solver checks the result against a list of these special functions and reports the integral as unsupported by name when it matches one.
Does the limit solver handle one-sided limits differently from two-sided ones?: Yes, but only for the blow-up case. A 0/0 indeterminate form is resolved with L'Hopital's rule, which gives the same answer regardless of direction, so left, right and two-sided all use it. For a limit that diverges to infinity, the solver numerically probes each side separately with shrinking offsets, since approaching from the left and right can genuinely disagree there.
What's the highest-order Taylor series I can generate?: The order field is capped at 12. That limit exists because the series is built by repeatedly taking symbolic derivatives one at a time rather than from a closed-form coefficient formula, so higher orders would mean noticeably slower computation for diminishing practical benefit in a browser-based visualizer.
Is any of this computation sent to a server?: No. The symbolic engine, nerdamer, runs entirely as JavaScript inside your browser tab, and every derivative, integral, limit or series expansion is computed there with no network request. You can confirm this by disconnecting from the internet after the page has loaded and continuing to use every operation normally.
Can it find second or higher-order derivatives?: The derivative panel offers the first and second derivative, and the second shows both differentiation passes as separate labelled steps. Higher orders are not a menu option, although the Taylor series panel differentiates repeatedly and prints the derivative values at the expansion point for the first few orders.
Does it show integration by parts or u-substitution steps?: Not in detail. For a product such as x^2*exp(x) the Method step names substitution or by-parts as the approach, then goes straight to nerdamer's antiderivative; the choice of u and dv and the intermediate integrals are not written out.
Can it graph the function or its derivative?: No, the solver only returns symbolic steps and results with no plot. To see a curve, its tangent lines or the shaded area under it, paste the same expression into the Graphing Calculator, which draws it and estimates roots, extrema and definite integrals numerically.
What input syntax does it accept?: Standard calculator notation: ^ for powers, * for multiplication, parentheses around function arguments, and names like sin, cos, sqrt, exp and log. Write the variable consistently, and use Infinity when a limit point is unbounded.

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Content reviewed 2026-09-27.