Calculus Calculator
Pick Derivative or Integral, type f(x) and read the answer. Derivative mode returns f'(x) as a formula and, if you give a point, the slope and tangent line there. Integral mode approximates the definite integral between two finite limits with the trapezoid, midpoint or Simpson rule. It does not do limits, series or exact antiderivatives.
Last updated November 4, 2026
Use x, + - * / ^, brackets, sqrt, ln, sin, cos, e^x. Write 2*x, not 2x. Trig is in radians.
A number or pi / e such as pi/2. Leave blank to skip.
Result
f'(x) = x^x*(ln(x) + 1)
f(2) = 4; f'(2) = 6.772588722
Tangent line: y = 6.772588722x - 9.545177444
Symbolic result. Trig in radians.
Show the work
Formula: Differentiate term by term with the sum, constant multiple, product, quotient, power and chain rules. Trig functions use radians.
- Read the function: f(x) = x^x.
- Logarithmic differentiation on x^x: (u^v)' = u^v*(v'*ln(u) + v*u'/u).
- Derivative of x on x: d/dx[x] = 1.
- Assemble the rules: f'(x) = x^x*(1*ln(x) + 1).
- Simplify: f'(x) = x^x*(ln(x) + 1).
- Evaluate at x = 2: f(2) = 4 and f'(2) = 6.772588722.
- Tangent line: y - 4 = 6.772588722(x - 2), which gives y = 6.772588722x - 9.545177444.
Result: f'(x) = x^x*(ln(x) + 1)
What each mode computes
Derivative: (u*v)' = u'*v + u*v'
Integral: integral of f from a to b ≈ h × bracket, h = (b - a) / n
Derivative mode applies the sum, constant multiple, product, quotient, power and chain rules to your expression and lists each rule it used. The result is exact, up to the modest simplification the calculator performs. A point a adds f(a), the slope f'(a) and the tangent line y - f(a) = f'(a)(x - a).
Integral mode is numerical. It splits the interval from a to b into n equal pieces of width h and combines the values of f at the pieces. Simpson with an error estimate runs twice, on n and on 2n pieces, and reports the finer value with a heuristic error estimate, about |S(2n) - S(n)| / 15 for smooth functions and adjusted when convergence is slower. Trig functions use radians in both modes.
See how we calculate for the formulas, sources and review schedule behind every calculator.
When to use the dedicated pages
This page is the quick entry point for both operations. Use the Derivative Calculator when you want the simplify switch and a page focused on rules and tangent lines. Use the Integral Calculator when you want to compare rules and see the n, 2n and 4n convergence table.
Errors and edge cases
A blank or unreadable f(x) is rejected with a message. If f or f' is undefined at the point a, no tangent line is given. Reversed limits are allowed and give the negative of the integral; equal limits give 0. A function that is undefined inside the interval, such as ln(x) across 0, is reported instead of approximated, and infinite limits are not supported.
How to use the Calculus Calculator
- Choose Derivative or Integral in the Operation menu.
- Type f(x) with x as the variable, for example x^x or sin(x). Write 2*x, not 2x.
- Derivative: optionally enter a point a. Integral: enter the lower and upper limits, n and the rule.
- Read the result, then open Show the work for the rules or the step-by-step sums.
Worked examples: d/dx x^x at x = 2, and sin(x) from 0 to pi
For x^x both base and exponent contain x, so logarithmic differentiation gives f'(x) = x^x*(ln(x) + 1). At x = 2 that is 4*(ln 2 + 1), about 6.772588722. The integral of sin(x) from 0 to pi is exactly 2; Simpson with n = 1000 agrees to many digits.
| Quantity | Result |
|---|---|
| f(x) = x^x, f'(x) | x^x*(ln(x) + 1) |
| f'(2) | 6.772588722 |
| Tangent line at x = 2 | y = 6.772588722x - 9.545177444 |
| Integral of sin(x), 0 to pi (Simpson, n = 1000) | ≈ 2 |
| Estimated error | about 6.8e-14 |
Frequently asked questions
Does the integral mode give an exact antiderivative?
No. It returns a numerical approximation of a definite integral between two finite limits, together with an error estimate for the Simpson option.
Are angles in degrees or radians?
Radians, in both modes. Type pi for angles, so a point of pi/2 means 90 degrees.
What happens if the function is undefined at my point or interval?
The calculator shows the engine message, such as f(x) not being defined at x = 0 for ln(x), instead of returning a wrong number.
Can it find limits, series or second derivatives?
No. It covers the first derivative of one function of x and numerical definite integrals only. For a second derivative, paste the result back in as f(x).