How to Find a Derivative

To find a derivative, identify the rule that matches your function's structure, then apply it to get an expression for the rate of change at any point. Most calculus problems reduce to three rules: the power rule, the product rule, and the chain rule.

What Is a Derivative?

A derivative tells you how fast a function is changing at any given input. Geometrically, it is the slope of the tangent line to the curve at that point. Practically, derivatives show up whenever something is moving or changing: speed is the derivative of position, acceleration is the derivative of speed, and marginal cost in economics is the derivative of total cost.

The notation d/dx means "derivative with respect to x." So d/dx of f(x), often written f'(x), gives you a new function whose output is the slope of f at each x. For a full formal treatment, see Wolfram MathWorld on the derivative.

If you want to check your work on a specific function, our derivative calculator will show the result with the rule applied.

The Power Rule

The power rule handles any term of the form x raised to a constant exponent. It is the workhorse of basic differentiation.

Bring the exponent down in front as a coefficient, then reduce the exponent by one. That is it. The rule works for positive integers, negative exponents, and fractions.

A few quick results worth memorizing: the derivative of x is 1 (since x = x^1, and 1 * x^0 = 1). The derivative of any constant is 0, because a constant has no rate of change. The derivative of x^3 is 3x^2. Each one follows the same pattern.

Worked Example 1: Power Rule on a Polynomial

Find the derivative of f(x) = 4x^3 + 5x^2 - 7x + 2.

Each term is handled independently. That is because differentiation is a linear operation: you can split a sum and differentiate piece by piece.

The Product Rule

When two functions are multiplied together, you cannot simply differentiate each and multiply the results. You need the product rule.

The pattern is: differentiate the first, leave the second alone, then add the first left alone times the derivative of the second. Some people remember it as "first times d-second plus second times d-first."

Worked Example 2: Product Rule

Find the derivative of h(x) = x^2 * (3x + 1).

You can verify this by expanding h(x) = 3x^3 + x^2 first, then differentiating directly. The answer is the same either way: 9x^2 + 2x.

The Chain Rule

The chain rule handles composite functions: situations where one function is plugged inside another. If y = f(g(x)), the derivative is f'(g(x)) times g'(x). In plain terms: differentiate the outer function, leave the inner function in place, then multiply by the derivative of the inner function.

A classic example: the derivative of (3x + 1)^5. The outer function is something raised to the 5th power; the inner function is 3x + 1. Differentiate the outer (bring down the 5, reduce the exponent): 5 * (3x + 1)^4. Then multiply by the derivative of the inner (which is 3): result is 15 * (3x + 1)^4. Short and clean once you see the structure.

When to Use Each Rule

The rule you reach for depends on how the function is built.

• Single term with a power? Power rule. Covers x^n, constants, and negative exponents like 1/x.
• Two functions multiplied together? Product rule. Covers things like x^2 * sin(x) or (x + 1)(x^2 - 3).
• A function inside another function? Chain rule. Covers things like sqrt(x^2 + 1) or e raised to a polynomial.
• Two functions divided? Quotient rule (a variant of the product rule): (f/g)' = (f'g - fg') / g^2.

Real problems often stack these. You might need the chain rule and the product rule on the same term. Work from the outside in, identify the outermost structure first, then handle what is inside.

Related reading: if you are working on linear functions, our guide on how to find the slope of a line covers the geometry that derivatives generalize.