Linearity Rules

Linearity Rules of Differentiation

Once you master the derivatives of basic functions, you need efficient rules for combining them. The linearity rules are the foundation: the constant multiple rule and the sum/difference rule.

The Two Core Rules

The constant multiple rule says constants "ride along"—they factor out untouched. The sum/difference rule says you differentiate term by term, preserving addition and subtraction. Together, these properties define linearity in calculus.

Why They Work

Both rules follow directly from the limit definition. For the constant multiple rule: $\frac{d}{dx}[c f(x)] = \lim_{h\to0} \frac{c f(x+h) - c f(x)}{h} = c \lim_{h\to0}\frac{f(x+h)-f(x)}{h} = c f'(x).$

Since limits are linear, the same logic applies to sums: the limit of a sum is the sum of limits.

Applying the Rules

In practice, differentiate entire polynomial-like expressions in one pass, term by term. For $f(x) = 3x^4 - 5x^2 + 7$:

• $\frac{d}{dx}[3x^4] = 3(4x^3) = 12x^3$
• $\frac{d}{dx}[-5x^2] = -5(2x) = -10x$
• $\frac{d}{dx}[7] = 0$

Result: $f'(x) = 12x^3 - 10x$.

Linearity also handles fractional and negative exponents. For $g(x) = 4\sqrt{x} - \dfrac{3}{x^2} + 2x$, rewrite as $g(x) = 4x^{1/2} - 3x^{-2} + 2x$, then differentiate: $g'(x) = 2x^{-1/2} + 6x^{-3} + 2 = \frac{2}{\sqrt{x}} + \frac{6}{x^3} + 2.$

Common Pitfalls

Don't multiply derivatives for sums: $\frac{d}{dx}[f(x)+g(x)] \neq f'(x) \cdot g'(x)$. Sums stay as sums.

Preserve the sign: $\frac{d}{dx}[f(x)-g(x)] = f'(x) - g'(x)$, not $f'(x)+g'(x)$.

Handle negatives carefully: $\frac{d}{dx}[-7x^3] = -21x^2$.

Mastering linearity is essential—every advanced rule (product, quotient, chain) builds on this foundation.