Calculus AB · Differentiation: Definition and Fundamental Properties · 14 min read · Updated 2026-05-10
Constant, sum, difference, and constant multiple rules — AP Calculus AB
AP Calculus AB · Differentiation: Definition and Fundamental Properties · 14 min read
1. The Constant Rule★☆☆☆☆⏱ 3 min
The constant rule is the simplest differentiation rule, applying to constant functions that output the same value for every input and have no dependence on $x$. It is derived directly from the limit definition of the derivative, and matches the intuitive fact that a horizontal line has a slope of 0 everywhere.
Exam tip: On the AP exam, constants like $\pi$, $e$, and arbitrary constants labeled $k$ in problems are still constants — always apply the constant rule to them, don't mistake them for variables.
2. The Constant Multiple Rule★★☆☆☆⏱ 3 min
The constant multiple rule extends differentiation to terms that have a constant coefficient multiplied by a non-constant function. It lets us pull constant coefficients out of derivatives, which is an essential step for term-by-term differentiation of polynomials.
Exam tip: Never forget that negative constants are still constants — keep the negative sign with the constant when you pull it out, don't accidentally drop it during simplification.
3. The Sum and Difference Rules★★☆☆☆⏱ 3 min
The sum and difference rules allow us to differentiate linear combinations of functions term-by-term, which is required to differentiate any polynomial. These rules follow directly from the limit property that the limit of a sum is the sum of the limits. The difference rule can also be derived by combining the sum rule with the constant multiple rule for $c=-1$.
Exam tip: Always check that you have differentiated every term in the function, including the final constant term — it is easy to forget it and accidentally carry it over into the final derivative.
4. Combined AP-Style Applications★★★☆☆⏱ 5 min
Common Pitfalls
Why: Students confuse the constant value itself with its derivative, forgetting the constant rule always outputs zero.
Why: Students only apply the derivative to the variable part and leave the original power of $x$ unchanged, forgetting to multiply the constant coefficient by the derivative of the variable part.
Why: Students confuse the constant $\pi$ with a variable and unnecessarily apply the product rule.
Why: Students forget to apply the constant rule to the constant term and incorrectly carry the constant into the derivative.
Why: Students fail to replace the final constant with zero, leaving it in the final derivative.