Calculus BC · Unit 2: Differentiation: Definition and Fundamental Properties · 14 min read · Updated 2026-05-11
Constant, sum, difference, and constant multiple rules — AP Calculus BC
AP Calculus BC · Unit 2: Differentiation: Definition and Fundamental Properties · 14 min read
1. The Constant Rule★☆☆☆☆⏱ 3 min
Exam tip: Any fixed symbol (like $\pi$, $e$, or $g$) is a constant, not a variable, unless the problem explicitly states it is a function of $x$. All fixed constants have derivative 0.
2. The Constant Multiple Rule★★☆☆☆⏱ 3 min
Exam tip: Always carry negative signs through when applying the constant multiple rule to negative constants. Dropping the negative is a common source of exam point loss.
3. The Sum and Difference Rules★★☆☆☆⏱ 3 min
Exam tip: When differentiating polynomials, work term by term explicitly to avoid missing terms or making constant errors. Even if you can do it in your head, writing it down reduces mistakes on exam day.
4. Combined AP-Style Worked Examples★★☆☆☆⏱ 5 min
Common Pitfalls
Why: $\pi$ is written with a Greek letter that looks like a variable, so students confuse it with the independent variable $x$.
Why: Students often focus on differentiating the function and forget to carry the negative sign from the original constant multiple.
Why: Students incorrectly extend the rule $\frac{d}{dx}[x] = 1$ to all constant terms, confusing the constant $c$ with the variable $x$.
Why: Students misapply signs when multiple terms are subtracted, leading to incorrect exponents or coefficients.
Why: Students confuse the constant multiple rule (scaling the entire function vertically) with the constant coefficient inside a composite function.