The Timpson Lyceum · Calculus 1 (2024)

Derivative at a Point & Derivative Functions - Lecture 8 (Calculus 1)

Overview

This lecture opens Chapter 3 by giving the derivative a precise definition. Intuitively, the derivative of a function at a point is just the slope of the function at that point — but a single point isn't enough to define a slope, so we build the slope of a curve as a limit of slopes of secant lines through two ever-closer points. That limit is the difference quotient limit, and it is the formal definition of the derivative. We then distinguish the derivative at a point (a single number) from the derivative function $f'(x)$ (a new function giving the slope everywhere), survey the many equivalent notations, and close by examining when a function fails to be differentiable — and why differentiability is a stronger condition than continuity.

Key Concepts

The derivative as a slope

The derivative of a function at a point is the slope of the function at that point. A horizontal tangent has derivative $0$ ; a vertical tangent has an undefined derivative.
You need two points to compute a slope. Through two points the slope is rise over run, $m = \dfrac{\Delta y}{\Delta x}$ , which gives the line's point–slope form $y - y_1 = m(x - x_1)$ .

From secant slopes to the tangent slope

To find the slope at a point $x_0$ , pick a second nearby point $x$ , draw the secant line through both, and use its slope as an approximation. As the second point slides toward $x_0$ , the approximation improves; the limit of these secant slopes is the exact slope at $x_0$ .
A derivative is therefore "the limit of a slope as the two points get arbitrarily close." Because the slope is a difference divided by a quotient, this expression is called the difference quotient.

The formal definition (equivalent forms)

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} = \lim_{z \to x} \frac{f(x) - f(z)}{x - z}$
The symbols $h$ and $\Delta x$ mean exactly the same thing (older texts favor $\Delta x$ ; newer texts favor $h$ ). Likewise, choosing the fixed point to be $x_0$ , $c$ , or $a$ is just relabeling — there is nothing special about the letter.
At a point the definition yields a single number (the slope there); as a function it yields a new function $f'(x)$ whose value at each input is the slope of $f$ there.

Notation for the derivative

$f'(x)$ , $\quad y'$ (when $y = f(x)$ ), $\quad \dfrac{dy}{dx}$ , $\quad \dfrac{d}{dx}f(x)$ .
The Leibniz form comes from replacing $\lim_{\Delta x \to 0}\dfrac{\Delta y}{\Delta x}$ with $\dfrac{dy}{dx}$ ; the last form treats $\dfrac{d}{dx}$ as an operator acting on $f$ . All denote the same object.

Application: velocity as a slope

If position is $f(t)$ , the average speed over an interval is displacement over time, $\dfrac{f(t_0 + \Delta t) - f(t_0)}{\Delta t}$ . Taking $\Delta t \to 0$ gives the instantaneous velocity — the slope of the position curve. So differentiating position gives velocity.

Where a derivative fails to exist

The difference quotient's denominator always tends to $0$ , so the entire technique is to algebraically cancel that factor before taking the limit.
A derivative does not exist where the function is not defined, where the tangent is vertical (slope blows up, as for $\sqrt{x}$ at $x = 0$ ), or at a corner/sharp point (as for $|x|$ at $x = 0$ ), where the left- and right-hand limits of the difference quotient disagree.
$f'$ has domain at most as large as $f$ 's domain: if $f$ isn't defined somewhere, it cannot have a slope there.

Differentiability and continuity

Differentiable means $f$ has a derivative at every point of its domain; on a closed interval $[a,b]$ this also requires one-sided derivatives at the endpoints to exist (be finite).
Continuity does not imply differentiability: $|x|$ is continuous at $0$ but has no derivative there.
Differentiability implies continuity: if $f$ is differentiable at $c$ , then $f$ is continuous at $c$ . Proof sketch: $\lim_{x \to c}\bigl(f(x) - f(c)\bigr) = \lim_{x \to c}\frac{f(x) - f(c)}{x - c}\,(x - c) = f'(c)\cdot 0 = 0,$ so $\lim_{x \to c} f(x) = f(c)$ .

Worked Example

Find the derivative of $f(x) = x^2$ from the definition, then use it to get the tangent line at $(2,4)$ .

Use $f'(x) = \lim_{z \to x}\dfrac{f(x) - f(z)}{x - z}$ , factoring the difference of squares to cancel the $x - z$ in the denominator:

\begin{aligned} f'(x) &= \lim_{z \to x} \frac{x^2 - z^2}{x - z} \\ &= \lim_{z \to x} \frac{(x - z)(x + z)}{x - z} \\ &= \lim_{z \to x} (x + z) \\ &= 2x. \end{aligned}

The slope at $x = 2$ is $f'(2) = 2(2) = 4$ . Using point–slope form through $(2,4)$ :

\begin{aligned} y - y_1 &= m\,(x - x_1) \\ y - 4 &= 4\,(x - 2) \\ y &= 4x - 4. \end{aligned}

A second example — $f(x) = \dfrac{1}{x}$ — shows the cancellation trick with a common denominator:

\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \\ &= \lim_{h \to 0} \frac{1}{h}\cdot\frac{x - (x+h)}{x(x+h)} \\ &= \lim_{h \to 0} \frac{-h}{h\,x(x+h)} \\ &= \lim_{h \to 0} \frac{-1}{x(x+h)} = -\frac{1}{x^2}. \end{aligned}

Summary

The derivative of $f$ at a point is the slope of $f$ there, defined as the limit of secant-line slopes (the difference quotient) as the two points coincide.
The three standard definition forms — $\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ , $\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$ , and $\lim_{z\to x}\frac{f(x)-f(z)}{x-z}$ — are logically identical; only the notation differs.
The derivative at a point is a single number; the derivative function $f'(x)$ ( $= y' = \frac{dy}{dx} = \frac{d}{dx}f$ ) returns the slope at every input.
Computing a derivative by definition always reduces to canceling the denominator's vanishing factor (factor, common denominator, or multiply by a conjugate) before taking the limit.
A derivative fails to exist at points where the function is undefined, where the tangent is vertical, or at corners — wherever the left- and right-hand difference quotients disagree.
Differentiability is strictly stronger than continuity: every differentiable function is continuous, but a continuous function (like $|x|$ ) need not be differentiable.