Derivative Calculator

The Derivative Calculator helps you find the derivative of a mathematical function with respect to a selected variable. Derivatives are an important part of calculus and are used to study rates of change, slopes, motion, optimization, and many real-world problems.

What Is a Derivative?

A derivative measures how quickly a function changes as its input changes. Geometrically, it represents the slope of a function at a particular point.

The derivative of a function f(x) is commonly written as:

f'(x) = d/dx [f(x)]

For example:

f(x) = x²

Its derivative is:

f'(x) = 2x

How to Use the Derivative Calculator

1. Enter the mathematical function you want to differentiate.
2. Select or enter the variable, such as x.
3. Click the calculate button.
4. The calculator will display the derivative of the function.

Derivative Formula

The derivative can be defined using a limit:

f'(x) = lim_{h → 0} [f(x + h) − f(x)] / h

This definition explains how the instantaneous rate of change of a function is determined.

Basic Derivative Rules

Several rules make differentiation easier:

• Constant Rule: d/dx(c) = 0
• Power Rule: d/dx(xⁿ) = nxⁿ⁻¹
• Sum Rule: The derivative of a sum is the sum of the derivatives.
• Product Rule: d/dx(fg) = f'g + fg'
• Quotient Rule: d/dx(f/g) = (gf' − fg') / g²
• Chain Rule: d/dx[f(g(x))] = f'(g(x))g'(x)

Derivative Example

Consider the function:

f(x) = 3x³ + 2x² − 5x + 7

Differentiate each term:

f'(x) = 9x² + 4x − 5

Therefore, the derivative of the function is 9x² + 4x − 5.

Why Are Derivatives Important?

Derivatives are used to understand how quantities change. They can help determine the slope of a curve, the velocity of a moving object, and the maximum or minimum values of a function.

Common applications include:

• Calculus and mathematical analysis
• Physics and motion
• Engineering
• Economics and finance
• Optimization problems
• Graph and curve analysis

First and Higher-Order Derivatives

The first derivative describes the rate of change of a function. Differentiating the result again gives the second derivative. Further differentiation produces higher-order derivatives.

For example, if:

f(x) = x³

Then:

f'(x) = 3x²

f''(x) = 6x

f'''(x) = 6

Frequently Asked Questions

What does a derivative tell you?

A derivative describes the instantaneous rate at which a function changes with respect to its variable. It can also represent the slope of a curve at a point.

What is the derivative of a constant?

The derivative of any constant is 0 because a constant does not change.

What is the power rule?

The power rule states that the derivative of xⁿ is nxⁿ⁻¹.

What is the difference between a derivative and a limit?

A derivative uses a limit to measure the instantaneous rate of change of a function. A limit describes the value a function approaches as its input approaches a particular value.

Related Calculators

If you are studying calculus, you can also use our Limit Calculator to evaluate limits. For integration problems, try our Integral Calculator. You can also explore our Math Calculators for more useful mathematical tools.

Conclusion

The Derivative Calculator provides a convenient way to differentiate mathematical functions and check calculus work. It is useful for students, teachers, engineers, and anyone working with rates of change, slopes, or mathematical models.