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Derivative Calculator

Calculate the derivative of common mathematical functions quickly and accurately. Enter a function in terms of x and get its derivative instantly.

Calculate a Derivative

Enter a mathematical function below.

Supported examples: x^2, x^3 + 2x, sin(x), cos(x), tan(x), e^x, ln(x), sqrt(x), 1/x
Try a common function:

Derivative Result

The derivative represents the instantaneous rate of change of a function.

Derivative Formula

The derivative of a function describes how quickly the function changes as its input changes. The derivative can be defined using the limit of the difference quotient.

Definition of the Derivative
f′(x) = lim [f(x + h) − f(x)] / h

In practical calculations, standard differentiation rules are usually used to find derivatives more efficiently.

Common Derivative Rules

Power Rule

d/dx (xⁿ) = nxⁿ⁻¹

Sine Rule

d/dx sin(x) = cos(x)

Cosine Rule

d/dx cos(x) = −sin(x)

Exponential Rule

d/dx eˣ = eˣ

Logarithm Rule

d/dx ln(x) = 1/x

Constant Rule

d/dx C = 0

How Does the Derivative Calculator Work?

The calculator recognizes common mathematical expressions and applies standard differentiation rules to determine the derivative with respect to x.

1

Enter a Function

Type a mathematical function such as x^2, sin(x), or e^x.

2

Apply Derivative Rules

The calculator identifies the expression and applies the appropriate differentiation rules.

3

View the Result

Your derivative is displayed immediately in a clear mathematical format.

Derivative Examples

Here are several common functions and their derivatives.

f(x) = x² f′(x) = 2x
f(x) = x³ f′(x) = 3x²
f(x) = sin(x) f′(x) = cos(x)
f(x) = cos(x) f′(x) = −sin(x)
f(x) = eˣ f′(x) = eˣ
f(x) = ln(x) f′(x) = 1/x

What Is a Derivative?

A derivative measures the instantaneous rate of change of a function. In geometry, it can be interpreted as the slope of the tangent line to a curve at a particular point.

Derivatives are fundamental in calculus and are used in mathematics, physics, engineering, economics, statistics, computer science, and many other fields.

Where Are Derivatives Used?

Frequently Asked Questions

What is a derivative?
A derivative measures the instantaneous rate of change of a function. It can also represent the slope of a tangent line to a curve.
What is the derivative of x²?
The derivative of x² is 2x, using the power rule.
What is the derivative of sin(x)?
The derivative of sin(x) is cos(x).
What is the derivative of cos(x)?
The derivative of cos(x) is −sin(x).
What is the derivative of a constant?
The derivative of a constant is zero because a constant does not change.
What is the power rule?
The power rule states that the derivative of xⁿ is nxⁿ⁻¹.