Fri Sep 11, 2026 Lecture (L08) Stewart Sect. 2.2 The Derivative as a Function, Part 2

Overview

Section 2.2 develops the derivative as a function by finding the derivative of f(t) = 1/√t directly from the limit definition, using a common denominator and conjugate to obtain f′(t) = −1/(2t^(3/2)). It then establishes that differentiability implies continuity and uses graph behavior to identify three failures of differentiability: discontinuities, a cusp, and a vertical tangent.

Key takeaways

Using the limit definition for f(t) = 1/√t requires combining fractions, multiplying by the conjugate, canceling h for h ≠ 0, and then evaluating the limit to get −1/(2t^(3/2)).
If f′(x) exists, the numerator f(x+h) − f(x) must approach zero as h→0; this gives lim(h→0) f(x+h) = f(x) and proves continuity at x.
The contrapositive of “differentiability implies continuity” is “discontinuity implies no derivative,” making discontinuity a sufficient test for failure of differentiability.
Continuity alone does not guarantee differentiability: a cusp can prevent a unique tangent line even when the function itself is continuous.
A vertical tangent also prevents a finite derivative because its slope is undefined, despite the tangent line itself being present.

Chapters

0:00 Section 2.2 Recap and a Definition-Based Derivative

Lecture 8 continues Stewart Section 2.2, following an earlier graphical and formula-based example involving a quadratic.
The class sets out to differentiate f(t) = 1/√t using the definition rather than shortcut rules.
The limit definition is emphasized as the required method when a problem explicitly requests it; faster derivative rules are introduced later.

2:05 Differentiate 1/√t by Rationalizing the Difference Quotient

The difference quotient is [1/√(t+h) − 1/√t]/h, which gives the indeterminate form 0/0 when h approaches zero.
Combining the two fractions over a common denominator prepares the numerator for multiplication by its conjugate.
The difference-of-squares identity reduces the numerator to −h; canceling h is valid because the limit takes h toward zero through nonzero values.
Substituting h = 0 after cancellation gives f′(t) = −1/(2t^(3/2)).

16:22 The Derivative Limit and Conditions for Differentiability

The derivative at x is defined as lim(h→0) [f(x+h) − f(x)]/h.
Because the denominator h approaches zero, a finite derivative limit requires the numerator to approach zero as well.
This requirement leads to lim(h→0) f(x+h) = f(x), the defining limit condition for continuity at x.

17:24 Why Differentiability at x Guarantees Continuity at x

If f′(x) exists, then the difference quotient has a limit, forcing f(x+h) − f(x) to approach zero.
Consequently, lim(h→0) f(x+h) = f(x), so f must be continuous at that x-value.
The implication is one-way: continuity is necessary for differentiability, but this result does not establish that every continuous function is differentiable.

24:59 Contrapositives: Turning the Continuity Result into a Test

For a conditional statement “if A, then B,” the converse “if B, then A” is not logically equivalent; the Cavaliers and Ferrari examples illustrate the distinction.
The contrapositive, “if not B, then not A,” is logically equivalent to the original conditional.
Applying that rule to differentiability yields: if f is not continuous at x, then f′(x) does not exist there.

30:11 Graph Examples: Three Discontinuities Prevent Derivatives

At graph location A, the left- and right-hand limits disagree, so the limit does not exist and the function is discontinuous.
At B, the limit exists but the function’s y-value is missing; at C, both exist but the limit and y-value differ.
Since the function is discontinuous at A, B, and C, the contrapositive guarantees that its derivative does not exist at those points.

32:36 Cusps and Vertical Tangents Also Block Differentiability

At D, the graph is continuous but has a cusp, so no single tangent line matches the graph’s direction from both sides.
At E, the tangent line is vertical; although a tangent line exists, its slope is undefined, so there is no derivative value.
The lecture identifies three general failure modes: discontinuity, no unique tangent line (as at a cusp), or a vertical tangent with undefined slope.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Barsamian's Math Videos.