The Sum and Difference Rules
Simple Explanation
The derivative of a sum (or difference) of terms is just the sum (or difference) of each term's own derivative: d/dx[f(x) ± g(x)] = f'(x) ± g'(x). This lets you differentiate a multi-term expression one term at a time.
Why Do We Need It?
Combined with the Power Rule, this lets you differentiate any polynomial in one pass, term by term.
Formula
The Sum and Difference Rules
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
The derivative of a sum (or difference) of functions is simply the sum (or difference) of their individual derivatives.
- f(x), g(x)
- — two differentiable functions
When to use it: Whenever a function is written as terms added or subtracted together — differentiate each term separately.
Worked Example
Differentiate a polynomial term by term
Differentiate f(x) = 3x² + 5x − 7.
Why Does This Work?
This follows directly from the limit definition: the limit of a sum equals the sum of the limits (a basic limit law), so [f(x+h)+g(x+h) − f(x) − g(x)]/h can be split into [f(x+h)−f(x)]/h + [g(x+h)−g(x)]/h, and taking the limit of each piece separately gives f'(x)+g'(x).
Real-Life Example
Differentiating a total cost function
A business's total cost function is built from several separate terms: materials cost, labor cost, and fixed overhead, each depending differently on production level x.
The sum rule lets each term be differentiated separately to find the marginal (total) cost function, rather than needing a single combined calculation.
Practice
Differentiate f(x) = 2x³ − 4x + 9, then evaluate f'(1).
EasyCommon mistake
Forgetting that the derivative of any constant term is exactly 0 — a constant term simply disappears entirely when differentiated, it does not carry through unchanged.
Quick Review
- d/dx[f(x) ± g(x)] = f'(x) ± g'(x).
- Differentiate a polynomial term by term.
- Every constant term differentiates to exactly 0.