Derivative, integral and limit calculator with steps

This online calculator finds derivatives, indefinite and definite integrals and limits of functions and shows the solution the way it is written by hand: every step names its rule and writes out the formula. The antiderivative is checked by differentiation, the derivative numerically and the limit with a table of values near the point, so the answer can be verified instead of taken on trust.

How to find a derivative, integral or limit

1

Choose what to find and type the function — for example, x^2 sin x or (1+1/x)^x. The preview under the field shows how it was read.

2

The answer and its check appear at once; set the limits for a definite integral and the point for a limit.

3

Under the answer come the steps and the graph: switch the level of detail, copy the solution or save a PDF.

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Enter a function and follow every rule of the solution

Function

Read as:
Examples:
Practice

A problem on the chosen topic with other numbers: solve it on paper and type your answer. Any correct form counts — 2 sin x cos x and sin 2x are the same, and the constant C of an antiderivative does not matter.

Notation
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Features

Derivatives up to the fifth order by the rules: sum, product, quotient, chain rule with the outer and inner function named, logarithmic differentiation for xˣ; partial derivatives, implicit functions F(x, y) = 0 and parametric curves x(t), y(t) with the second derivative.
Integrals: basic formulas, linear substitution, u-substitution, integration by parts (including the cyclic ∫eˣ sin x dx), partial fractions, power-reducing formulas, the substitution t = tan(x/2); definite integrals by the Fundamental Theorem, improper ones through a limit.
Limits: 0/0, ∞/∞, ∞ − ∞, 0·∞, 1^∞ — factoring, conjugates, equivalent infinitesimals, sin x / x and the limit that defines e, L'Hôpital's rule with an applicability check, one-sided limits.
Every answer is checked before it is shown: a derivative against a numerical derivative at 20 points, an antiderivative by differentiation, a definite integral by quadrature, a limit with values near the point.
Curve sketching: tangent and normal lines, increasing and decreasing intervals, extrema, concavity, and the absolute maximum and minimum on a closed interval, written out step by step.
Taylor and Maclaurin polynomials up to degree 10 with a table of derivatives at the point and a Peano remainder; the solution copies as text and LaTeX, saves as a PDF and is shared by link; practice problems by topic check your own answer.

What it solves

  • Derivatives: x^2 sin x, (x+1)/(x-1), sin(x^2), e^(2x) cos 3x, x^x, arctan(1/x), at a point and up to order 5
  • Indefinite integrals: x e^x, 1/(x^2-5x+6), sin^3 x, x/(x^2+1), sqrt(1-x^2), 1/(2+cos x)
  • Definite and improper integrals: sin x from 0 to pi, 1/x^2 from 1 to ∞, ln x from 0 to 1
  • Limits: sin(3x)/x as x → 0, (1+1/x)^x as x → ∞, (sqrt(x+1)-1)/x, one-sided 1/x as x → 0
  • Curve sketching: tangent line at a point, extrema, absolute maximum and minimum on an interval
  • Implicit and parametric: x^2 + y^2 = 25, x^3 + y^3 = 3xy, x = cos t, y = sin t; Taylor series: e^x, ln(1+x), sin x around a point

How to type a function

Powers use ^ or ²: x^2, x^(1/3), e^(2x). The multiplication sign can be left out: 2x, x sin x, (x+1)(x-1). Functions work with or without parentheses: sin x, sin 2x = sin(2x), sin^2 x, tan x, cot, arcsin, arctan, sinh, cosh, ln x, log_2 x, log(2, x), sqrt(x) or √x, cbrt(x), |x|. Constants are pi and e, infinity is ∞ or inf. A whole problem can be pasted: ∫ x^2 dx, ∫_0^1 x^2 dx, d/dx (x^2), lim x→0 sin(x)/x. Other letters (a, b, k) are treated as constants.

How this differs from a calculator that only gives the answer

Any computer algebra system gives the answer, but the grade is for the solution: which rule was applied at each step. Here the steps are not a retelling of the answer but a chain of equalities by the textbook rules — the chain rule names the outer and inner function, integration by parts writes out u and dv, and a limit names its indeterminate form and how it is resolved. Every answer is also checked by an independent numerical computation: if the symbolic answer and the check disagree, the tool says so instead of presenting an unchecked answer as correct.

Frequently asked questions

Why is the answer ln|x| and not ln x?

Because 1/x is defined for negative x too, and ln x is not. The derivative of ln|x| is 1/x on both halves of the axis, so the antiderivative of 1/x is ln|x| + C. Writing ln x would lose half of the domain. Where the expression inside the logarithm is always positive — x² + 1, eˣ + 1 — the absolute value is not written, and the tool leaves it out too.

My answer differs from the textbook. Which one is right?

Possibly both. An antiderivative is defined up to a constant: sin²x/2 and −cos 2x/4 differ by exactly 1/4 and have the same derivative. Different forms of a derivative are equal too: 2 sin x cos x = sin 2x. To check, differentiate your answer or type it into the practice box: answers there are compared by their values, not by their letters.

When can L'Hôpital's rule not be used?

When there is no 0/0 or ∞/∞ form — then the rule gives a wrong answer. And when the limit of the ratio of derivatives does not exist: for (x + sin x)/x as x → ∞ the derivatives give (1 + cos x)/1, which oscillates, although the limit itself is 1 — it is found by dividing term by term. The tool checks both conditions and says plainly when the rule does not apply.

Why can the integral of e^(−x²) not be found?

Its antiderivative cannot be written with elementary functions — this is a proven fact (Liouville's theorem), not a weakness of the calculator. The same holds for sin x / x, eˣ/x, 1/ln x and sin x². Special functions are introduced for them — the error function erf, the sine integral Si and others. A definite integral of such a function still exists and is computed numerically: from −∞ to ∞ the integral of e^(−x²) equals √π.

How do I find the absolute maximum of a function on a closed interval?

Find the derivative, solve f′(x) = 0, keep the roots that lie in the interval, and evaluate the function at the endpoints and at those points — the largest value is the answer. An extremum inside the interval does not have to be the largest value: the maximum may be at an endpoint. Set the interval in the curve sketching box and the solution is written out in these steps.

Why does an indefinite integral end with + C while a definite one gives a number?

An indefinite integral is all antiderivatives at once, and they differ by a constant: x² and x² + 5 have the same derivative 2x. That is why + C is added. In a definite integral the Fundamental Theorem subtracts the value at the lower limit from the value at the upper limit, the C cancels, and a number remains.

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