# How do you solve a calculus problem?

Derivatives, integrals and definite integrals of a function, each checked numerically.

- Page: https://www.acalculator.org/math/calculus-calculator
- JSON spec: https://www.acalculator.org/math/calculus-calculator.json
- Version: b37bfafdb541

## Default answer

Example with the default inputs (Function f(x) 4/(1 + x^2), Find Integral): An antiderivative of 4/(1 + x^2) is 4 atan(x) + C.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x) | The function, in one letter. |
| op | Find | What to find. |
| a | Lower limit a | A number, such as 0. |
| b | Upper limit b | A number, such as pi. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| derivative | Derivative | The derivative with respect to the letter. |
| value | Definite integral | The signed area under f from a to b. |
| antiderivative | Antiderivative | A function whose derivative is f, plus a constant C. |
| exact | Exact value | The definite integral, exactly. |

## Method

A computer algebra system finds each answer; it shows only after a numeric check.

## Assumptions

- One variable letter (x if none); radians; ln is the natural logarithm.
- An answer that fails its check is not shown.

## Worked examples

1. f = 4/(1 + x^2), op = integral gives antiderivative = 4 atan(x) + C. Source: OpenStax Calculus Vol. 1, 4.10, Ex. 4.52(c). https://openstax.org/books/calculus-volume-1/pages/4-10-antiderivatives.
2. f = t^2 - 4, op = definite, a = -2, b = 2 gives value = -10.666667, exact = -32/3.
3. f = 5x^2/(4x + 3), op = derivative gives derivative = 10x (2x + 3)/(4x + 3)^2.

## FAQ

### What can this calculus calculator do?

Three things for a function of one variable: the derivative (the rate of change), an antiderivative (the indefinite integral, + C), and a definite integral from a to b (a signed area, exactly and as a decimal). For limits use the limit calculator, and for a Taylor polynomial the Taylor series calculator.

### How are derivatives and integrals related?

They undo each other. By the fundamental theorem of calculus, if F′ = f then the definite integral of f from a to b is F(b) − F(a). That is also how the page checks an antiderivative: it differentiates it back numerically and compares with f.

### Why is there a + C?

The derivative of a constant is 0, so if F is an antiderivative of f, so is F + C for every constant C. 4 atan(x) + C covers every antiderivative of 4/(1 + x²). A definite integral is one number, because C cancels in F(b) − F(a).

### How is each answer checked?

A derivative is compared with a numeric difference quotient of f at 20 points; an antiderivative is differentiated numerically at 20 points and compared with f; and a definite integral is compared with numeric integration of f. An answer that fails is never shown.

### Can I use t instead of x?

Yes. Use any one lowercase letter as the variable, such as t² − 4, and the derivative or integral is taken with respect to it. Two different letters give no answer. The letter e is Euler’s number.

### What does "No verified answer" mean?

The algebra found no formula, its answer failed the numeric check, or the work took over 3 seconds. Some functions, such as eˣ/x, have no antiderivative in elementary functions; the page may then give one with a special function such as Ei(x), or no answer.

## Sources

- OpenStax, Calculus Volume 1, section 3.3 Differentiation Rules: https://openstax.org/books/calculus-volume-1/pages/3-3-differentiation-rules
- OpenStax, Calculus Volume 1, section 4.10 Antiderivatives: https://openstax.org/books/calculus-volume-1/pages/4-10-antiderivatives
- OpenStax, Calculus Volume 1, section 5.3 The Fundamental Theorem of Calculus: https://openstax.org/books/calculus-volume-1/pages/5-3-the-fundamental-theorem-of-calculus