How to Find the Derivative: Rules, Formulas, and Worked Examples

Derivatives are the foundation of calculus. They measure how a function changes as its input changes, giving us the instantaneous rate of change at any point. Whether you are studying physics, engineering, economics, or pure mathematics, mastering differentiation is essential.

This tutorial covers the four key differentiation rules with worked examples, so you can confidently find the derivative of any function you encounter.

What Is a Derivative?

The derivative of a function at a point is the slope of the tangent line to the graph at that point. Formally, it is defined as:

In practice, we rarely use the limit definition directly. Instead, we apply a set of rules that make differentiation fast and systematic.

Rule 1: The Power Rule

The power rule is the most frequently used rule in differentiation. For any function of the form , the derivative is:

Example 1: Simple Power Rule

Find the derivative of .

Example 2: Polynomial

Find the derivative of .

Apply the power rule term by term (and note that the derivative of a constant is zero):

Example 3: Negative and Fractional Exponents

Find the derivative of .

For square roots, rewrite as a fractional exponent. The derivative of is:

Rule 2: The Product Rule

When you need to differentiate the product of two functions, use the product rule:

Example 4: Product Rule

Find the derivative of .

Let and , so and .

Rule 3: The Quotient Rule

For the ratio of two functions:

Example 5: Quotient Rule

Find the derivative of .

Let and , so and .

Rule 4: The Chain Rule

The chain rule is used to differentiate composite functions, i.e. functions nested inside other functions:

Example 6: Chain Rule

Find the derivative of .

The outer function is and the inner function is .

Example 7: Chain Rule with Trigonometric Function

Find the derivative of .

Common Derivatives Reference Table

Here are the standard derivatives you should know:

• (constant)

Tips for Success

1. Always simplify first. Rewrite roots as fractional exponents and fractions as negative exponents before differentiating.
2. Identify which rule to use. Is it a product? A quotient? A composite function? Choose the correct rule before starting.
3. Combine rules when needed. Many real functions require two or more rules. For example, differentiating uses both the product rule and the chain rule.
4. Check your answer. Use our calculator to verify your work and see step-by-step solutions.

Try It Yourself

Practise finding derivatives with our free Derivative Calculator. It shows the full solution with each differentiation step, so you can follow along and learn from every problem.