Math & Science Calculator

Calculus Calculator

Use this calculus calculator to explore derivatives, definite integrals, and limits from one simple workspace. Enter a function such as x^2 + 3x, choose the operation, and get a calculated result with the formula and method explained. This calculus calculator is designed for learning as well as checking homework: it shows what was evaluated, explains the key idea, and helps you understand how each answer is obtained.

Calculate a derivative, integral, or limit

Use x as the variable. Enter multiplication explicitly when possible, such as 3*x. Trigonometric functions use radians.

Supports standard arithmetic, powers, parentheses, constants and functions such as sin, cos, tan, sqrt, log, exp, abs.

Derivative results use symbolic differentiation where supported. Definite integrals and limits are numerical estimates, so check the displayed method and domain restrictions.

How to use the calculus calculator

  1. Choose Derivative, Definite integral, or Limit.
  2. Type your function using x, for example x^2, sin(x), or sqrt(x).
  3. Enter the point or bounds requested for that operation.
  4. Select Calculate and review the result and method notes.

What is calculus?

Calculus is the branch of mathematics concerned with change and accumulation. Differential calculus studies rates of change through derivatives, while integral calculus studies accumulated quantities through integrals. Limits describe the value a function approaches near a chosen input and provide the foundation for both derivatives and integrals.

Derivative formula

The derivative of a function describes its instantaneous rate of change. From the limit definition,

f′(x) = limh→0 [f(x + h) − f(x)] / h

For example, if f(x) = x^2 + 3x, then f′(x) = 2x + 3. At x = 2, the derivative is 7. Symbolic output depends on the expression being supported by the math engine.

Definite integrals and numerical estimation

A definite integral measures signed accumulation over an interval:

∫ab f(x) dx

This tool estimates the integral with composite Simpson’s rule, which approximates the area using parabolic segments. It is useful for many smooth functions, but the result is an approximation and can be less reliable near discontinuities, singularities, or rapidly changing functions.

Understanding limits

A limit describes the value a function approaches as x gets close to a specified number. A two-sided limit exists only when the left-hand and right-hand behavior agree. The numerical estimate shown here samples values increasingly close to the target; it cannot prove a limit exists and may struggle with oscillating functions, discontinuities, or removable singularities.

Examples to try

  • Derivative: x^3 at x = 2 gives 12.
  • Definite integral: x^2 from 0 to 3 is approximately 9.
  • Limit: (x^2 - 1)/(x - 1) as x approaches 1 approaches 2, although the original expression is undefined at exactly x = 1.

Accuracy and limitations

Calculations run in your browser using a mathematical expression engine. Derivatives are symbolic when the engine can differentiate the entered expression. Integrals and limits are numerical estimates, not a computer algebra proof. Use parentheses to make the intended order of operations clear, avoid undefined input values, and compare important results with a textbook or another trusted method.

Frequently asked questions

What can I calculate with this calculus calculator?

This calculus calculator handles symbolic derivatives, numerical definite integrals, and numerical estimates of left-hand, right-hand, and two-sided limits. Choose the operation that matches your problem, then enter the function and required values.

Can I use this as an online calculus solver for derivatives?

Yes. Choose Derivative to differentiate a supported function symbolically. The result includes the derivative expression and its value at your selected x-coordinate. For example, the derivative of x^2 + 3*x is 2*x + 3.

Does the calculus solver find indefinite integrals?

No. The integral option estimates a definite integral between the lower and upper bounds you enter. It does not currently return a symbolic antiderivative for an indefinite integral.

Are calculus calculator answers exact?

Derivative expressions are produced symbolically when the entered function is supported, although evaluating the derivative at a point gives a numerical value. Definite integrals and limits are numerical estimates, not exact symbolic solutions or mathematical proofs.

How do I enter a function correctly?

Use x for the variable, ^ for powers, and parentheses to make the order clear. Write multiplication explicitly, such as 3*x. Common functions include sin(x), cos(x), sqrt(x), log(x), exp(x), and abs(x). Trigonometric functions use radians.

Why might a limit result say no matching finite estimate?

The left-hand and right-hand values may approach different numbers, the function may be undefined near the target, or numerical sampling may be unreliable for that expression. Compare one-sided estimates and remember that numerical sampling cannot prove a limit exists.

Can I change the function and calculate again without resetting?

Yes. Edit the function, point, bounds, or operation and press Calculate again. The result panel is refreshed for each new calculation; Reset is only needed when you want to restore the example values.