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Module 1.1 · Topic 4

Differentiability

⏱ 18 min★ Up to 90 XP

The Setup

Two roads.

You are driving. At the marked point — what direction are you facing?

Road A — Smooth highway

Road B — Sharp V-shaped turn

Cell Growth

Population curves.

Smooth growth

Can we define the instantaneous growth rate?

The Derivative

What happens as Q approaches P?

Average Rate of Change

—

Interactive Simulation

Watch the secant become a tangent.

Secant

Slope

4.02

Distance Between Points

h ≈ 1.02

← 0 (tangent)

large (secant) →

Formal Definition

f′(x)=

limh → 0

f(x+h) − f(x)h

Take average change.

f(x+h) − f(x) measures the rise. Dividing by h gives the slope over the interval.

Shrink the interval.

Let h → 0. The interval collapses to a single point.

Observe what remains.

The slope that survives — that is f′(x).

Where it breaks

Three failure modes.

Failure Type 1

Corner

slope = −1

slope = +1

Which slope should we choose?

No unique answer.

Not differentiable.

Failure Type 2

Cusp

Failure Type 3

Vertical Tangent

What is the slope?

Infinite.

Derivative breaks down.

Differentiable

↓

Continuous

Ask

If something is differentiable, must it be continuous?

↑ pHTemperature →

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