Defining the derivative and using derivative notation — AP Calculus AB
1. The Derivative at a Point ★★☆☆☆ ⏱ 4 min
We derive the derivative at a point from the precalculus concept of the slope of a secant line between two points $(a, f(a))$ and $(a+h, f(a+h))$, where $h$ is the step size between the points. As $h \to 0$, the points get closer together, and the secant slope approaches the tangent slope (the derivative).
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
An equivalent alternate form, most commonly used for recognition problems rather than manual calculation, replaces step size $h$ with $x$ approaching $a$:
f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
Exam tip: If an AP question explicitly asks you to use the limit definition to find a derivative, you earn zero credit for only using a shortcut differentiation rule; you must show the full limit calculation to earn points.
2. The Derivative as a Function and Derivative Notation ★★☆☆☆ ⏱ 5 min
Once we can calculate the derivative at any single point $x=a$, we can treat the derivative as a function itself, where the input is any $x$ where the derivative exists, and the output is the derivative at that $x$.
Two standard notations are used interchangeably on the AP exam:
- **Prime notation**: For $y = f(x)$, the derivative is written $f'(x)$ (read "f prime of x") or $y'$. For the derivative at $x=a$, this becomes $f'(a)$ or $y'(a)$.
- **Leibniz notation**: For $y = f(x)$, the derivative is written $\frac{dy}{dx}$ (read "dee y dee x") or $\frac{d}{dx}[f(x)]$, where $\frac{d}{dx}$ denotes the operation of taking the derivative. For the derivative at $x=a$, this is written $\frac{dy}{dx}\bigg|_{x=a}$, with the evaluation bar after the derivative expression.
Exam tip: When writing the derivative evaluated at a point in Leibniz notation, always place the evaluation bar after the derivative expression; misplacing the bar will be marked incorrect on FRQs.
3. Recognizing Derivatives from Limit Expressions ★★★☆☆ ⏱ 3 min
A common AP exam question gives you an unsimplified limit of a difference quotient and asks you to either identify what derivative it represents or evaluate the limit using derivative rules, instead of calculating the limit from scratch. This tests whether you understand the structure of the derivative definition, not just how to compute with it.
To solve these problems, match the given limit to the structure of one of the two derivative definitions:
- $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = f'(a)$
- $\lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f'(a)$
Once you correctly identify $f(x)$ and $a$, you can evaluate the derivative using shortcut rules to get the value of the limit, without computing the limit by hand.
Exam tip: Always check that $f(a)$ matches the constant term in the numerator; common MCQ distractors use the wrong constant term to test if you match all parts of the definition correctly.
4. Concept Check (AP-Style) ★★★☆☆ ⏱ 2 min
Common Pitfalls
Why: Students confuse the order of operations for limits, thinking substitution is always the first step.
Why: Students match the constant in the numerator to the point, ignoring the denominator structure.
Why: Students mix up the two types of derivative problems after practicing both.
Why: Students confuse the existence of the limit of the original function with the existence of the derivative.
Why: Students misapply fraction notation to derivative evaluation.