Sum and Difference Rules
Textbook Reference
| Primary source | OpenStax Calculus Volume 1, Section 3.3: “Differentiation Rules” |
| Book URL | https://openstax.org/details/books/calculus-volume-1 |
Freely available and openly licensed.
Key idea
The rate of change of a sum is the sum of the rates of change. This is not obvious from daily intuition, but it follows directly from the limit law that allows limits to be distributed over sums. The sum rule is what makes differentiation a “linear” operation: you can differentiate term by term.
Prerequisite Check
Quick Reference
\[ \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \] \[ \frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x) \]
Differentiation distributes over sums and differences.
Key Concepts
1. Proof from the Definition
Let $h(x) = f(x) + g(x)$. Then: \[ h'(x) = \lim_{s \to 0} \frac{[f(x+s)+g(x+s)] - [f(x)+g(x)]}{s} = \lim_{s \to 0} \left[\frac{f(x+s)-f(x)}{s} + \frac{g(x+s)-g(x)}{s}\right]. \] The limit of the sum equals the sum of the limits (since both limits exist): \[ h'(x) = f'(x) + g'(x). \]
The difference rule follows by applying the constant multiple rule with $c = -1$ to $g$.
2. Differentiating Polynomials Term by Term
Combining the sum rule, constant multiple rule, and power rule:
Example 1. $f(x) = 3x^4 - 5x^2 + 2x - 7$.
$f'(x) = 3(4x^3) - 5(2x) + 2(1) - 0 = 12x^3 - 10x + 2$.
Example 2. $g(x) = x^5 + x^3 - x$.
$g'(x) = 5x^4 + 3x^2 - 1$.
Example 3. $h(x) = \dfrac{2x^3 - 6x}{4}$.
Simplify first: $h(x) = \dfrac{x^3}{2} - \dfrac{3x}{2}$.
$h'(x) = \dfrac{3x^2}{2} - \dfrac{3}{2}$.
3. What the Sum Rule Does NOT Allow
The sum rule applies when both terms are differentiable. It does not apply when the two functions are multiplied (that requires the product rule) or composed (that requires the chain rule).
$\dfrac{d}{dx}[f(x) \cdot g(x)] \neq f'(x) \cdot g'(x)$. This is a common error.
Common Errors
| Error | Example | Correction |
|---|---|---|
| Applying sum rule to products | “$\frac{d}{dx}[(x^2)(x^3)] = 2x \cdot 3x^2$” | Products need the product rule; or simplify: $x^5$, then $5x^4$ |
| Missing a term | Forgetting to differentiate the constant $-7$ | $\frac{d}{dx}[-7] = 0$; write 0 explicitly, then drop it |
Leveled Practice
Level 1 -- Polynomial Differentiation
Problem 1. Differentiate: $y = x^4 - 3x^3 + 5x - 2$.
Show answer
$y' = 4x^3 - 9x^2 + 5$.
Problem 2. Find $g'(x)$ for $g(x) = 7x^3 + 4x^2 - x + 1$.
Show answer
$g'(x) = 21x^2 + 8x - 1$.
Level 2 -- Using the Derivative
Problem 3. For $f(x) = x^3 - 6x^2 + 9x$, find $f'(x)$ and find all values of $x$ where $f'(x) = 0$.
Show answer
$f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x-1)(x-3)$.
$f'(x) = 0$ at $x = 1$ and $x = 3$.
Problem 4. Find the equation of the tangent line to $y = 2x^3 - 3x + 1$ at $x = 1$.
Show answer
$y(1) = 2 - 3 + 1 = 0$. Point: $(1, 0)$.
$y'(x) = 6x^2 - 3$; $y'(1) = 3$. Slope: 3.
Tangent: $y = 3(x - 1) = 3x - 3$.
Level 3 -- Simplify then Differentiate
Problem 5. Find $f'(x)$ for $f(x) = \dfrac{x^4 - 3x^2 + 2}{x}$.
Show answer
Divide each term by $x$: $f(x) = x^3 - 3x + \dfrac{2}{x} = x^3 - 3x + 2x^{-1}$.
$f'(x) = 3x^2 - 3 - 2x^{-2} = 3x^2 - 3 - \dfrac{2}{x^2}$.
Common Misconceptions
the derivative of a product is the sum of the derivatives. The sum rule says $(f + g)' = f' + g'$, which is correct. Students sometimes extend this to products: $(fg)' = f'g'$. This is wrong. The derivative of $x \cdot x = x^2$ is $2x$, but $1 \cdot 1 = 1 \neq 2x$. Products require the product rule.
differentiating term by term applies to any algebraic combination. The linearity of differentiation (sum and constant-multiple rules) applies only to sums, differences, and constant multiples. It does not extend to products, quotients, or compositions. The expression $(3x^2)(5x + 1)$ cannot be differentiated by taking the derivative of each factor separately and adding; the product rule is required.
Mastery Checklist
Mental Model
Differentiation is linear: like taking a derivative “distributes” over a sum just as multiplication distributes over addition in arithmetic. The sum rule says you can differentiate term by term and add the results. This is the fundamental reason polynomials are easy to differentiate: they are sums of monomials, and each monomial’s derivative is found by the power rule.
Connections
Within MATH161
- Polynomial differentiation: Every polynomial derivative uses the sum rule. All derivative rules produce results that are then combined with the sum rule.
- Integration: $\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx$ is the same linearity property applied to antidifferentiation.
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