Real Analysis 34 | Differentiability [dark version]

TL;DR

Differentiability at x0 is equivalent to the existence of a limit of secant slopes as x approaches x0.

Briefing Cornell Notes

Briefing

Differentiability at a point is fundamentally about whether a function can be approximated locally by a straight line, and that straight-line “best fit” is determined by a limit of secant slopes. The key idea is pointwise: at a fixed point x0, the function has a derivative if the slopes of lines through (x0, f(x0)) and (x, f(x)) settle toward a single number as x approaches x0. When that limit exists, the function’s local behavior is captured by a linearization—its tangent line—whose slope is f′(x0). This matters because it turns geometric slope information into a precise analytic criterion, enabling rigorous work with rates of change and local modeling.

The discussion starts by contrasting a “sharp bend” graph (not smooth) with a parabola-like curve (smooth), then reframes smoothness in terms of differentiability, which behaves like continuity: it is a property checked at individual points. To see where the derivative comes from, the transcript introduces linear functions as the simplest model. A linear function g(x) = Mx + C has constant slope M, and rewriting it around a chosen point x0 as g(x) = M(x − x0) + g(x0) makes clear how the constant term shifts to match the function’s value at x0. From this representation, the slope can be computed via the difference quotient (g(x) − g(x0)) / (x − x0), which is valid for x ≠ x0.

For a general function f, the same quotient is used to approximate slope locally. Pick x0 and another nearby point x; the line through the two points is a secant, and its slope is (f(x) − f(x0)) / (x − x0). As x moves closer and closer to x0, the secant lines approach a tangent line. The derivative f′(x0) is defined as the limit of these secant slopes as x → x0, provided the limit exists. The transcript emphasizes that the “differential quotient” notation is just a limit expression rather than an actual fraction.

To formalize differentiability, the transcript generalizes from functions on all real numbers to functions defined on an interval I (or an open set). Differentiability at x0 means the relevant limit exists using points from I near x0. It also introduces a helper function ΔF,x0 (a difference-quotient function) that packages the quotient as a function of x; then differentiability becomes equivalent to continuity of this Δ-function at x0. Finally, the pointwise nature is stressed: differentiability at one point does not automatically imply differentiability at other points, so each x0 must be checked separately. The payoff is that later results can rely on this rigorous local linear approximation framework.

Cornell Notes

Differentiability at a point x0 means the function f can be locally approximated by a line whose slope is the limit of secant slopes. For x ≠ x0, the secant slope through (x0, f(x0)) and (x, f(x)) is (f(x) − f(x0)) / (x − x0). If this expression approaches a single number as x → x0, that number is the derivative f′(x0). The derivative is often written using limit notation (and sometimes as DF/DX at x0), but it is not a literal fraction—just a standard way to denote the limit. Differentiability is pointwise: it can hold at x0 without holding at other points.

Why does the derivative come from secant lines rather than directly from the curve’s shape?

The construction starts with a linear model: a line’s slope is constant and can be computed by a difference quotient. For a general function f, there is no single slope everywhere, so the slope is approximated using a secant line through two points (x0, f(x0)) and (x, f(x)). The secant slope is (f(x) − f(x0)) / (x − x0). The derivative is then defined by taking the limit of these secant slopes as x approaches x0, turning the “best local line” idea into a precise criterion.

What exactly is f′(x0), and how is it defined?

f′(x0) is the number obtained when the secant slope converges as x → x0. Concretely, f′(x0) exists if the limit of (f(x) − f(x0)) / (x − x0) exists as x approaches x0 (with x staying in the domain near x0). That limiting value is the slope of the tangent line at x0, giving the local linearization of f.

How does the transcript connect differentiability to continuity?

It introduces a difference-quotient function ΔF,x0(x) that equals (f(x) − f(x0)) / (x − x0) for x in the domain (x ≠ x0). Then differentiability at x0 is tied to whether this Δ-function has a limit behavior at x0. In the transcript’s framing, the limit exists exactly when this Δ-function is continuous at x0. This lets continuity intuition replace repeatedly writing the limit in later definitions.

Why does the domain matter (interval I vs. all real numbers)?

The limit process requires points arbitrarily close to x0. So the function must be defined on a set I that contains enough neighboring points around x0—such as an interval (open or closed) where x can approach x0 from within the domain. If the domain doesn’t provide points near x0, the limit defining the derivative may not be meaningful or may fail to exist.

What does “pointwise property” mean for differentiability?

Differentiability is checked at a specific point x0. Even if f is differentiable at x0, that does not guarantee differentiability at another point x1. Each point requires its own secant-slope limit test, because the local behavior of f can change from one location to another.

Review Questions

Compute the derivative definitionally: what limit must exist for f′(x0) to be defined?
Explain how a tangent line arises as a limit of secant lines, and identify the formula for the secant slope.
Why can differentiability at one point not imply differentiability at another point?

Key Points

1
Differentiability at x0 is equivalent to the existence of a limit of secant slopes as x approaches x0.
2
The secant slope through (x0, f(x0)) and (x, f(x)) is (f(x) − f(x0)) / (x − x0) for x ≠ x0.
3
When the secant-slope limit exists, its value is the derivative f′(x0), which is the slope of the tangent line.
4
Differentiability is pointwise: it must be verified separately at each point x0.
5
The derivative can be denoted using f′(x0) or alternative limit-based notations like DF/DX evaluated at x0.
6
Differentiability can be reframed as continuity of a difference-quotient function ΔF,x0 at x0.
7
The domain must include points near x0 so the limit process can be taken within the set I.

Highlights

A derivative is defined by the limit of slopes of secant lines, not by a single geometric measurement.

The tangent line is the limiting linear approximation as x approaches x0.

The “differential quotient” is a limit expression written in quotient-like notation, not an actual fraction to be simplified.

Differentiability behaves like continuity: it can be expressed through the continuity of a difference-quotient function at x0.

Differentiability at one point gives no automatic information about other points.

Topics

Differentiability
Linearization
Secant and Tangent
Difference Quotient
Derivative Definition