# Online Derivative and Definite Integral Calculator

> Free online calculus calculator: first and second derivative at a point with the tangent line, or a definite integral by adaptive Simpson with an error bound.

- URL: https://qilingtools.com/en/math/calculus
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Type a function and get derivatives and integrals. The derivative view gives f(x₀), the first and second derivatives, the tangent and normal lines and the slope angle; the integral view uses adaptive Simpson’s rule and reports an error estimate and the average value on the interval.

## How to use

1. Choose “Numerical derivative” or “Definite integral”
2. Enter f(x), for example x^2*sin(x), 1/(1+x^2) or e^(-x^2)
3. Enter the point x₀, or the limits a and b — results update as you type

## Features

- Central differences with Richardson extrapolation, accurate to about 1e-10
- Second derivative, tangent line, normal line and slope angle included
- Adaptive Simpson’s rule refines the sampling where the function changes fastest
- Error estimate and average value, with a warning on discontinuities
- Same syntax as the graphing tool, so you can plot the same expression

## FAQ

### Is this a symbolic or numerical result?

Numerical: it does not return a derivative formula but the value at the point you choose and the integral over the interval you choose — accurate enough for homework and engineering estimates.

### How accurate is the derivative?

Central differences with Richardson extrapolation typically give an error around 1e-10 for smooth functions; results are shown to 10 significant digits.

### What if the interval contains a discontinuity?

The tool warns that the integrand is infinite or discontinuous. Split the interval at the problem point and integrate each piece separately.

### How do I write the function?

Exactly as in the graphing tool: ^ for powers, implicit multiplication (2x), functions sin cos tan ln lg sqrt abs exp and constants pi, e.

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