Short on time? This page is the recap. For explanations, Figure 1 and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: f′(a) = lim (h → 0) (f(a + h) − f(a))/h; h → 0⁻ means from the left, h → 0⁺ from the right.
Recap
- f′(a) exists only if both one-sided limits of the difference quotient exist, are finite and are equal.
- Differentiable at a ⇒ continuous at a. Reason: f(x) − f(a) = [difference quotient] × (x − a) → f′(a) × 0 = 0.
- Contrapositive: a hole, jump or asymptote at a means f′(a) does not exist.
- Domain: if f(a) is undefined, f′(a) is undefined, even if the graph “looks” like a line through the hole.
- Converse is false: a continuous function can have a corner, cusp or vertical tangent.
Key relationships
| Feature at x = a | Continuous? | Difference quotient | f′(a)? |
|---|---|---|---|
| Smooth graph | Yes | One finite limit | Exists |
| Corner | Yes | Finite one-sided limits that differ | Does not exist |
| Cusp | Yes | −∞ on one side, +∞ on the other | Does not exist |
| Vertical tangent | Yes | Unbounded, same sign both sides | Does not exist |
| Hole, jump or asymptote | No | Not needed: use the theorem | Does not exist |
Assumptions behind the method
- One-sided quotients must use the actual f(a), from whichever piece includes a.
- For a piecewise function, check continuity first, then compare the one-sided limits.
- Absolute values only cause corners where the inside changes sign.
Mistakes to avoid
- Reversing the theorem: “continuous, so differentiable” or “not differentiable, so not continuous”.
- Matching slopes but ignoring a jump between the pieces.
- Writing f′(a) = ∞ for a vertical tangent. Say it does not exist.
- “It’s sharp” with no limits. Give the two one-sided values.
- Trusting a calculator’s numerical derivative at a corner; it averages the two sides.
- Assuming a formula with |x| has a corner. Check: x|x| is differentiable at 0.
Quick self-check
- f′(5) exists. Must f be continuous at 5? (Yes: differentiability implies continuity.)
- f is continuous at 0. Must f′(0) exist? (No: for example, a corner.)
- Is |2x − 6| differentiable at x = 3? (No: the one-sided limits of the quotient are −2 and 2.)
Next: practice questions.