Common Mistakes
Calculus BC · AP · Differentiation — Definition and Fundamental Properties
Focus
Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
This topic is assessed on the AP Exam.
Frequent mistakes
### 1. Limit notation misuse
What it looks/sounds like: ‘=’ is used where ‘approaches’ is required.
Why it happens: Limits and function values are treated as identical.
How to fix: Reserve ‘=’ for when the limit equals the function value at a point.
### 2. Average vs instantaneous rate
What it looks/sounds like: f′(a) is confused with average rate on [a, b].
Why it happens: Slope of secant and tangent are not distinguished.
How to fix: Ask: point (instantaneous) or interval (average)?
### 3. Missing chain-rule factor
What it looks/sounds like: Derivative of f(g(x)) omits g′(x).
Why it happens: Only the outer rule is applied.
How to fix: Circle inner and outer; multiply by the inner derivative.
### 4. Correct work, wrong question
What it looks/sounds like: The algebra is fine but answers a different quantity than the stem asks for.
Why it happens: Attention locks onto the first operation instead of the final ask.
How to fix: Underline the final question; after solving, verify the boxed answer matches it.
### 5. Notation and justification gaps
What it looks/sounds like: Missing units, limit language, or ‘because’ statements AP graders expect.
Why it happens: Notation is treated as decoration rather than part of the score.
How to fix: Add one justification phrase per major step (definition, theorem, or rule).
How to avoid them
1. State derivative meaning with units in context. 2. Show which differentiation rule applies each step. 3. Check answers with a quick estimate or second method.
Quick self-check
- [ ] Can compute a derivative with correct notation
- [ ] Can interpret f′ in context
- [ ] Can apply product/quotient/chain correctly