Math

Find the Derivative

How to Find the Derivative of Any Function

Finding the derivative tells you how fast a function is changing at any point. Enter your function below and this calculator will find the derivative for you, showing which rule applies at each step. No sign-up required.

Derivative Calculator Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Power Rule

Differentiate a polynomial using the power rule.

Key values: f(x) = x^3 - 2x^2 + x · 1st derivative

Chain Rule

Differentiate a composite function with sin(x^2).

Key values: f(x) = sin(x^2) · Chain rule applied

Product Rule with Tangent

Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.

Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0

Second Derivative

Compute the second derivative to analyze concavity.

Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative

Documentation

This calculator is also known as Find the Derivative.

Read the complete guide

What Is a Derivative?

A derivative measures how much a function's output changes when its input changes by a tiny amount. If f(x) = x^2, the derivative f'(x) = 2x tells you that at x = 3, the function is increasing at a rate of 6 units per unit.

Which Rule Do I Use?

Use this decision tree to pick the right rule:

Is it a constant? Use the constant rule (derivative is 0)
Is it x^n? Use the power rule
Is it a sum or difference? Differentiate each term separately
Is it a product f*g? Use the product rule
Is it a quotient f/g? Use the quotient rule
Is it a composite f(g(x))? Use the chain rule

Examples

Simple Polynomial

Find the derivative of f(x) = x^3 - 2x + 1.

Apply the power rule: 3x^2 - 2. The constant 1 vanishes.

Key takeaway: Polynomials are differentiated term by term using the power rule.

Trigonometric Function

Find the derivative of f(x) = sin(x).

The derivative of sin(x) is cos(x). This is a standard result to memorize.

Key takeaway: Memorize the basic trig derivatives: sin -> cos, cos -> -sin.

Tips for Finding Derivatives

These tips will help you find derivatives more efficiently:

Start by identifying the outermost operation in your expression
Practice the power rule until it becomes automatic
Remember that the chain rule applies whenever you see a function of a function

Frequently Asked Questions about Find the Derivative

How do I find the derivative step by step?

Enter your function in the calculator. It will automatically identify which differentiation rules apply and show each step. For manual practice: (1) identify the type of expression, (2) apply the appropriate rule, (3) simplify.

What does the derivative tell me about a function?

The derivative tells you the rate of change. Where the derivative is positive, the function is increasing. Where it's negative, the function is decreasing. Where the derivative is zero, the function has a critical point (potential maximum or minimum).