2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Constant and Constant Multiple Rules

The derivative of a constant is always 0, since a constant function never changes. The constant multiple rule says d/dx[k·f(x)] = k·f'(x) — a constant factor just carries through the derivative unchanged. These rules let you quickly differentiate terms like 5 or 7x³ without going back to the limit definition.

Sum and Difference Rules

The sum rule: d/dx[f(x) + g(x)] = f'(x) + g'(x). The difference rule: d/dx[f(x) − g(x)] = f'(x) − g'(x). You can differentiate each term of a polynomial or combined function separately, then add or subtract the results — no need to expand or use the limit definition each time.

Putting it together with the Power Rule

Combine these rules with the power rule (d/dx[xⁿ] = n·xⁿ⁻¹) to differentiate any polynomial term-by-term. For example, d/dx[3x⁴ − 2x + 7] = 12x³ − 2 + 0 = 12x³ − 2. This term-by-term approach is the backbone of fast, reliable differentiation.

Key terms

Constant Rule

d/dx[c] = 0 for any constant c.

Constant Multiple Rule

d/dx[k·f(x)] = k·f'(x); constants factor out of the derivative.

Sum Rule

d/dx[f(x) + g(x)] = f'(x) + g'(x).

Difference Rule

d/dx[f(x) − g(x)] = f'(x) − g'(x).

Term-by-term differentiation

For polynomials, differentiate each term separately using the power, constant, sum, and difference rules, then combine.

Try an AP-style question

d/dx (4x³ − 5x + 9) = ?

1. 12x² − 5 + 9
2. 4x² − 5
3. 12x³ − 5x
4. 12x² − 5