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Calculus Calculator

Quick answer

This calculator computes derivatives and integrals online with our free calculus calculator.

  • Free, no sign-up, works on any device.
  • More calculators and guides on Alathasiba.

Figures are estimates from this calculator’s standard formula — adjust the inputs in the tool below for your exact number.

How the Calculus Calculator Works

Derivative: f'(x) = lim(h->0) [f(x+h) - f(x)] / h | Integral: integral f(x)dx = F(x) + C

Differentiation finds the instantaneous rate of change (slope) of a function. Integration is the reverse operation, finding the area under a curve or the antiderivative. These fundamental concepts are essential in physics, engineering, economics, and all quantitative sciences.

  1. 1

    Select Operation

    Choose derivative or integral

  2. 2

    Enter Function

    Input the mathematical function (e.g., x^2, sin(x), e^x)

  3. 3

    Apply Rules

    The calculator applies appropriate differentiation or integration rules

  4. 4

    View Solution

    The result displays with detailed solution steps

Use Cases

Physics and Motion

Differentiate position to get velocity, velocity to get acceleration

Economics

Calculate marginal cost, marginal revenue, and optimization problems

Engineering

Compute areas, volumes, and centers of mass

Statistics

Find probability density functions and expected values

Tips

  • 1

    Power rule for derivatives: d/dx(x^n) = n x^(n-1)

  • 2

    Chain rule: d/dx[f(g(x))] = f'(g(x)) x g'(x)

  • 3

    Always add constant of integration C for indefinite integrals

  • 4

    Integral of 1/x is ln|x| + C (absolute value is important)

Common Mistakes

  • Forgetting the constant of integration C - indefinite integrals have infinitely many solutions

  • Applying power rule when n = -1 - integral of x^(-1) is ln|x| + C, not x^0/0

  • Forgetting the chain rule - d/dx[sin(2x)] = 2cos(2x), not just cos(2x)

  • Confusing d/dx with partial derivatives - d/dx is for single variable functions

FAQs

What is the difference between differentiation and integration?

Differentiation finds the rate of change (slope). Integration is the reverse, finding the area under a curve or recovering the original function.

What is the power rule for derivatives?

d/dx(x^n) = n x^(n-1). Example: d/dx(x^3) = 3x^2. Works for any real exponent.

Why do we add + C in integration?

Because the derivative of any constant is zero, infinitely many functions have the same derivative. C represents this arbitrary constant.

What is definite vs indefinite integration?

Indefinite integral gives a function F(x) + C. Definite integral from a to b gives a number: F(b) - F(a), representing the area.

What is the chain rule?

d/dx[f(g(x))] = f'(g(x)) x g'(x). Used for composite functions. Example: d/dx[sin(x^2)] = cos(x^2) x 2x.

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