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Master Derivative Rules

Complete guide to derivative rules with free interactive practice tools. Learn the power rule, constant rule, and sum rule with step-by-step explanations and master the fundamental techniques of differentiation.

What are Derivative Rules?

Understanding the fundamental rules for finding derivatives

Derivative rules are formulas that allow us to find the derivative of functions efficiently. The derivative measures how a function changes at any given point, representing the instantaneous rate of change. The three fundamental rules—power rule, constant rule, and sum rule—form the foundation for differentiating polynomial functions and many other types of expressions.

xⁿ

Power Rule

Learn how to differentiate functions with powers using the power rule: d/dx(xⁿ) = nxⁿ⁻¹, the most commonly used derivative rule.

Practice Power Rule →
Σ

Sum Rule

Master the sum rule which states that the derivative of a sum is the sum of derivatives: d/dx[f(x) + g(x)] = f'(x) + g'(x).

Practice Sum Rule →
C

Constant Rule

Understand that the derivative of any constant is zero, and learn how constants interact with other functions in differentiation.

Derivative Rules and Properties

Master the fundamental rules for finding derivatives

Power Rule

The derivative of x raised to a power n is n times x raised to the power (n-1). This is the most fundamental derivative rule.

\frac{d}{dx}(xⁿ) = nxⁿ⁻¹
\frac{d}{dx}(x^{3}) = 3x^{2}
\frac{d}{dx}(x^{5}) = 5x^{4}
\frac{d}{dx}(x) = x^{0} = 1

Constant Rule

The derivative of any constant is zero. Constants don't change, so their rate of change is always zero.

\frac{d}{dx}(C) = 0
\frac{d}{dx}(5) = 0
\frac{d}{dx}(-3) = 0
\frac{d}{dx}(100) = 0

Constant Multiple Rule

When a function is multiplied by a constant, the derivative is the constant times the derivative of the function.

\frac{d}{dx[cf}(x)] = \frac{c d}{dx[f}(x)]
\frac{d}{dx}(3x^{2}) = \frac{3 d}{dx}(x^{2}) = 3(2x) = 6x
\frac{d}{dx}(-2x^{3}) = \frac{- 2 d}{dx}(x^{3}) = -2(3x^{2}) = -6x^{2}

Sum Rule

The derivative of a sum (or difference) is the sum (or difference) of the derivatives. This allows us to differentiate polynomials term by term.

\frac{d}{dx[f}(x) + g(x)] = f'(x) + g'(x)
\frac{d}{dx}(x^{3} + 2x) = \frac{d}{dx}(x^{3}) + \frac{d}{dx}(2x)
= 3x^{2} + 2
\frac{d}{dx}(4x^{2} - 3x + 1) = 8x - 3

Negative Exponents

The power rule works for negative exponents too. Remember that x⁻ⁿ = 1/xⁿ, so d/dx(x⁻ⁿ) = -nx⁻ⁿ⁻¹.

\frac{d}{dx}(x⁻²) = -2x⁻³ = \frac{- 2}{x^{3}}
\frac{d}{dx}(x⁻¹) = -x⁻² = \frac{- 1}{x^{2}}

Fractional Powers

The power rule also applies to fractional exponents. For x^(1/n), the derivative is (1/n)x^(1/n - 1).

\frac{d}{dx}(x^(\frac{1}{2})) = (\frac{1}{2})x^(\frac{- 1}{2}) = \frac{1}{2√x}
\frac{d}{dx}(x^(\frac{1}{3})) = (\frac{1}{3})x^(\frac{- 2}{3}) = \frac{1}{3x^(\frac{2}{3}})

Derivative Examples

See worked examples of different derivative types

Basic Examples

\frac{d}{dx}(x^{3})3x^{2}
\frac{d}{dx}(5)0
\frac{d}{dx}(2x^{2})4x
\frac{d}{dx}(x^{3} + 2x)3x^{2} + 2
\frac{d}{dx}(x⁻²)-2x⁻³

Practice Difficulty Levels

Understand what each difficulty level covers in our practice generator

Easy

Perfect for beginners learning the basics of differentiation.

  • • Power rule only
  • • Constants
  • • Single-term expressions

Medium

Requires applying multiple rules and understanding how they combine.

  • • Sum rule
  • • Multiple terms
  • • Coefficients

Hard

Advanced differentiation with complex expressions.

  • • Negative exponents
  • • Fractional powers
  • • Mixed expressions (algebraic complexity)

Why Derivative Rules Matter

Derivative rules are essential tools in calculus that allow us to find rates of change efficiently. They form the foundation for understanding how functions behave, enabling us to solve optimization problems, analyze motion, and model real-world phenomena. Without these rules, finding derivatives would require going back to the limit definition every time, which would be extremely time-consuming.

Applications

Derivative rules are used extensively in physics to find velocity and acceleration, in economics to determine marginal costs and revenues, in engineering to optimize designs, and in biology to model population growth rates. They're fundamental to understanding any situation where rates of change are important.

Start Practicing Derivative Rules

Practice derivatives with our infinite generator and master differentiation techniques

d/dx

Infinite Derivative Practice

Practice finding derivatives with our infinite generator. Get instant feedback and master derivative rules including power rule, constant rule, and sum rule.

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