MATH1961 · Mathematics 1a (advanced)
Differentiation
This is the proof-heavy core of the calculus stream. In the Advanced unit the derivative is not ‘the slope you remember from school’ — it is a limit of difference quotients, and you are expected to know why it forces continuity and how the rules are proved. Lectures use the Carathéodory reformulation (a clean product form) because it makes the product- and chain-rule proofs short. The centrepiece is the Mean Value Theorem family: Rolle’s theorem, the MVT itself, and the Cauchy MVT, with the monotonicity and extremum corollaries that follow, and L’Hôpital’s rule proved from the Cauchy MVT. Every ‘prove that…’ item the exam loves — differentiable implies continuous, the product rule from the definition, Rolle → MVT — is foregrounded here.
What this chapter covers
- 01The derivative as a limit of difference quotients — the secant → tangent picture
- 02The Carathéodory form, and proving differentiable ⇒ continuous
- 03The four rules, and how lectures prove the product rule from the definition
- 04Rolle’s theorem → the Mean Value Theorem → the Cauchy MVT
- 05Monotonicity & extremum corollaries; L’Hôpital’s rule from the Cauchy MVT
Worked example: prove differentiable implies continuous
- +1State the goal. Continuity at a means limx→a f(x) = f(a), equivalently limx→a [f(x) − f(a)] = 0. We are given that f′(a) = limx→a (f(x) − f(a))/(x − a) exists.
- +1Set up the algebraic identity. For x ≠ a write f(x) − f(a) = [(f(x) − f(a))/(x − a)] · (x − a) — multiply and divide by (x − a).
- +1Take the limit of each factor. The first factor → f′(a) (given, finite); the second factor (x − a) → 0. By the product law for limits the product → f′(a) · 0 = 0.
- +1Conclude. So limx→a [f(x) − f(a)] = 0, i.e. limx→a f(x) = f(a): f is continuous at a. The converse is false — |x| is continuous but not differentiable at 0. ■
Key terms
- Derivative
- f′(a) = limh→0 (f(a+h) − f(a))/h, the limit of secant slopes as the second point approaches a. If the limit exists the secants converge to one tangent line of that slope; no limit means no tangent (a corner, cusp or vertical tangent), so the function is not differentiable there.
- Carathéodory form
- An equivalent definition: f is differentiable at a iff f(x) = f(a) + φ(x)(x − a) for some φ continuous at a, with φ(a) = f′(a). It trades a fragile quotient for a clean product, which is what makes the product- and chain-rule proofs short.
- Rolle's theorem
- If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f′(c) = 0 for some c in (a, b). It is the base case from which the Mean Value Theorem is proved by tilting the picture.
- Mean Value Theorem (MVT)
- For f continuous on [a, b] and differentiable on (a, b), there is c in (a, b) with f′(c) = (f(b) − f(a))/(b − a): some instantaneous slope equals the average slope. It powers the monotonicity test (f′ > 0 ⇒ increasing), Taylor’s remainder and L’Hôpital.
- L'Hopital's rule
- For an indeterminate 0/0 or ∞/∞ quotient, lim f/g = lim f′/g′ when the right side exists. In MATH1961 it is proved from the Cauchy MVT, and you must verify the indeterminate form before differentiating — applying it to a non-indeterminate form is a marked error.
Differentiation FAQ
Why does the Advanced unit insist the derivative is a limit, not just a slope formula?
Because the proofs depend on it. From the limit definition (or the equivalent Carathéodory form) you can prove that differentiability forces continuity, and you can derive the product, quotient and chain rules rather than memorising them. Mainstream calculus uses the rules; MATH1961 asks you to justify them, and that is where the marks separate.
What is the Carathéodory form for and why do lectures use it?
It rewrites differentiability as f(x) = f(a) + φ(x)(x − a) with φ continuous at a and φ(a) = f′(a). This is a clean product instead of a fragile quotient, so the product-rule and chain-rule proofs become short, gap-free arguments. It is the form the lecturer’s proofs are built on, so learn it alongside the standard definition.
How does the whole MVT family fit together?
Rolle’s theorem is the base case (equal endpoints force a flat tangent). Tilt the picture and you get the Mean Value Theorem (some slope equals the average slope). Generalise to two functions and you get the Cauchy MVT, from which L’Hôpital’s rule follows. So the chain is Rolle → MVT → Cauchy MVT → L’Hôpital — and the monotonicity and extremum tests are MVT corollaries.
What is the most common L'Hopital mistake?
Applying it without confirming the indeterminate form. You may only use L’Hôpital on 0/0 or ∞/∞, and you may only iterate while each new quotient is still indeterminate. Substitute first, name the form, then differentiate top and bottom separately — that one discipline line is where the mark is won.
Exam move
Learn both forms of the derivative — the difference-quotient limit and the Carathéodory product — because the exam’s ‘prove from the definition’ items use them. Rehearse the named proofs as a set: differentiable ⇒ continuous, the product rule from the definition, and the Rolle → MVT → Cauchy MVT chain; write each as numbered steps you can reproduce on a blank page. For L’Hôpital, build the habit of substitute → name the form → differentiate, and never iterate past an indeterminate form. Pair every theorem with its picture (the secant collapsing to the tangent, the tilted Rolle diagram) so you can both state and justify it under time.
