Basic Derivative Rules

Differentiation is the process of finding the derivative of a function — a measure of how that function's output changes as its input changes. Rather than returning to the limit definition every time, mathematicians have developed a set of shortcut rules that let us differentiate common expressions quickly and reliably. The most fundamental of these are the basic derivative rules: the constant rule, the scalar multiple rule, the sum rule, and the difference rule. These four rules form the bedrock of all differentiation. Together they allow you to differentiate any polynomial, and they combine with other rules (product, quotient, chain) to handle far more complex functions. Understanding why each rule works — not just how to apply it — gives you a powerful intuition that will carry you through all of calculus. In this article we derive and apply each rule carefully, with attention to notation and meaning. By the end you should be able to differentiate expressions like f(x) = 7x⁴ − 3x² + 5x − 9 in a few lines, understanding every step you take.

Key takeaways
  • The Constant Rule
  • The Power Rule (prerequisite context)
  • The Scalar Multiple Rule (Constant Multiple Rule)
  • The Sum Rule
  • The Difference Rule
  • Combining All Four Rules

The Constant Rule

If f(x) = c for some constant c, then f'(x) = 0. Geometrically, the graph of a constant function is a horizontal line with slope 0 everywhere. Using the limit definition: f'(x) = lim[h→0] (c − c)/h = lim[h→0] 0/h = 0. So the derivative of any constant is zero.

The Power Rule (prerequisite context)

Although a separate rule in its own right, the power rule d/dx[xⁿ] = nxⁿ⁻¹ underpins polynomial differentiation and is assumed here. For example, d/dx[x³] = 3x². The basic rules in this article describe how to handle combinations of such terms.

The Scalar Multiple Rule (Constant Multiple Rule)

If f(x) = c·g(x) where c is a constant and g is differentiable, then f'(x) = c·g'(x). You may 'factor out' a constant coefficient before differentiating. Proof from limits: d/dx[c·g(x)] = lim[h→0] (c·g(x+h) − c·g(x))/h = c · lim[h→0] (g(x+h) − g(x))/h = c·g'(x). For example, d/dx[5x⁴] = 5 · 4x³ = 20x³.

The Sum Rule

If h(x) = f(x) + g(x) where both f and g are differentiable, then h'(x) = f'(x) + g'(x). In words: the derivative of a sum is the sum of the derivatives. Proof: lim[h→0] [(f(x+h)+g(x+h)) − (f(x)+g(x))]/h = lim[h→0] (f(x+h)−f(x))/h + lim[h→0] (g(x+h)−g(x))/h = f'(x) + g'(x), using the limit sum law.

The Difference Rule

If h(x) = f(x) − g(x), then h'(x) = f'(x) − g'(x). This follows immediately from the sum rule combined with the scalar multiple rule: f(x) − g(x) = f(x) + (−1)·g(x), so the derivative is f'(x) + (−1)·g'(x) = f'(x) − g'(x). For example, d/dx[x⁵ − 3x²] = 5x⁴ − 6x.

Combining All Four Rules

In practice all four rules are used simultaneously on polynomials. Given p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, we differentiate term by term: each constant vanishes, each coefficient factors out, and the power rule handles each xᵏ. The result is p'(x) = n·aₙxⁿ⁻¹ + (n−1)·aₙ₋₁xⁿ⁻² + … + a₁.

The Key Equation

ddx[c⋅f(x)±g(x)]=c⋅f′(x)±g′(x)\frac{d}{dx}\bigl[c\cdot f(x) \pm g(x)\bigr] = c\cdot f'(x) \pm g'(x)
  • \frac{d}{dx} — Differential operator (dimensionless): Instructs us to differentiate with respect to the variable x.
  • c — Scalar constant (dimensionless): A real number that multiplies a function; it factors out unchanged during differentiation.
  • f'(x) — Derivative of f (depends on context): The instantaneous rate of change of f with respect to x.
  • g'(x) — Derivative of g (depends on context): The instantaneous rate of change of g with respect to x, added or subtracted according to the ± sign.

Worked Example

Problem. Find the derivative of f(x) = 4x³ − 7x² + 2x − 9.

f′(x)=12x2−14x+2f'(x) = 12x^2 - 14x + 2

Answer. f'(x) = 12x² − 14x + 2

Figure
Diagram
Derivative of f(x) = x² − 2x at various pointsxyf(x)=x²−2xx=3, slope=f'(3)=4x=1, slope=f'(1)=01203−2−15208Tangent lines show the derivative (slope) at x=1 and x=3. Here f'(x)=2x−2.
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently developed calculus in the 1670s and invented the notation d/dx and the integral sign ∫ that we still use today. His systematic rules for differentiating sums, products, and quotients laid the groundwork for the standard rule-based approach to differentiation taught in every calculus course. Leibniz also invented an early mechanical calculator called the Stepped Reckoner that could multiply and divide — a remarkable feat of engineering for the 17th century.

  • The Power Rule — The sum, difference, and scalar rules are most useful when combined with the power rule to differentiate polynomials term by term.
  • The Limit Definition of the Derivative — Each basic rule is proved rigorously using the limit definition, so understanding limits deepens your understanding of why the rules work.
  • The Product Rule — Once the basic rules are mastered, the product rule extends differentiation to products of two functions.
  • The Chain Rule — The chain rule is needed for composite functions and builds directly on the scalar multiple and basic rules.
  • Continuity — Differentiability implies continuity; understanding continuous functions helps you identify where derivative rules can be applied.