The Timpson Lyceum · Real Analysis (2020)

Lecture 13 (Real Analysis 2020)

Overview

This lecture introduces the derivative — the third of the four big ideas of calculus (limit, continuity, derivative, integral). Intuitively, the derivative is the slope of a function at a single point, made precise as a special kind of limit. We start from the familiar middle-school slope of a line (rise over run), generalize it to the slope of a curve by letting two points slide together, and arrive at the formal definition as the limit of a difference quotient. We then build the basic machinery: the tangent line, the fact that differentiability implies continuity, and the standard rules for differentiating sums, products, reciprocals, quotients, and power functions.

Key Concepts

Slope of a line (motivation). For a straight line through two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ , the slope is the rise over the run: $m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}.$ The slope is the same no matter which two points you pick — the defining feature of a line.

The difference quotient. For a general function and a fixed point $x_0$ , the slope of the secant line through $(x_0, f(x_0))$ and $(x, f(x))$ is $\frac{f(x_0) - f(x)}{x_0 - x}.$ This expression is called the difference quotient (a difference of values in a quotient). As $x$ moves closer to $x_0$ , the secant slope better approximates the slope at $x_0$ .

Neighborhood. An open interval $(a,b)$ is the set of all points strictly between $a$ and $b$ . An open interval containing $x_0$ is called a neighborhood of $x_0$ . We typically write $I$ for a neighborhood of $x_0$ .

Definition of the derivative. Let $I$ be a neighborhood of $x_0$ and $f : I \to \mathbb{R}$ . We say $f$ is differentiable at $x_0$ provided the limit $f'(x_0) = \lim_{x \to x_0} \frac{f(x_0) - f(x)}{x_0 - x}$ exists, in which case this limit is the derivative of $f$ at $x_0$ . (Because it is a limit, $x$ never actually equals $x_0$ , so the denominator is never zero in the limiting process.) If $f$ is differentiable at every point of $I$ , we call $f$ differentiable.

When the derivative fails to exist. A function has no derivative where the slope is not well defined — at a sharp corner (e.g. $|x|$ at $0$ ), or anywhere it is discontinuous (no derivative without continuity).

The tangent line. If $f$ is differentiable at $x_0$ , the tangent line to $f$ at $(x_0, f(x_0))$ is $y = f(x_0) + f'(x_0)\,(x - x_0).$ This mirrors the point–slope form of a line; $y = mx + b$ is the special case $x_0 = 0$ . Once the derivative exists at a point, this tangent line is unique.

Differentiability implies continuity. If $f$ is differentiable at $x_0$ , then $f$ is continuous at $x_0$ (i.e. $\lim_{x \to x_0} f(x) = f(x_0)$ ). The converse fails: $|x|$ is continuous everywhere but not differentiable at $0$ , and there even exist functions continuous everywhere yet nowhere differentiable.

Differentiation rules. For functions differentiable at $x_0$ :

Constant slope of a line: if $f(x) = mx + b$ , then $f'(x) = m$ .
Power rule: if $f(x) = x^{a}$ for an integer $a$ , then $f'(x) = a\,x^{a-1}$ (proved first for natural numbers, then extended to all integers).
Constant multiple: if $f(x) = a\,x^{n}$ , then $f'(x) = a\,n\,x^{n-1}$ .
Sum rule: $(f + g)'(x_0) = f'(x_0) + g'(x_0)$ .
Product rule: $(fg)'(x_0) = f'(x_0)\,g(x_0) + g'(x_0)\,f(x_0)$ — not the product of the derivatives.
Reciprocal rule: if $g(x_0) \neq 0$ , then $\left(\tfrac{1}{g}\right)'(x_0) = \dfrac{-g'(x_0)}{\,g(x_0)^2\,}$ .
Quotient rule: if $g(x_0) \neq 0$ , then $\left(\tfrac{f}{g}\right)'(x_0) = \dfrac{g(x_0)\,f'(x_0) - f(x_0)\,g'(x_0)}{g(x_0)^2}$ ("low d-high minus high d-low, square the bottom").
Consequence: every polynomial is differentiable everywhere, and every rational function is differentiable wherever its denominator is nonzero.

Worked Example

Prove from the definition that if $f(x) = x^2$ , then $f'(x_0) = 2x_0$ . The key trick is to factor the numerator so the troublesome $x_0 - x$ in the denominator cancels, after which the limit is found by direct substitution.

\begin{aligned} f'(x_0) &= \lim_{x \to x_0} \frac{f(x_0) - f(x)}{x_0 - x} \\ &= \lim_{x \to x_0} \frac{x_0^{\,2} - x^{2}}{x_0 - x} \\ &= \lim_{x \to x_0} \frac{(x_0 - x)(x_0 + x)}{x_0 - x} \\ &= \lim_{x \to x_0} (x_0 + x) \\ &= x_0 + x_0 = 2x_0. \end{aligned}

As a check on the product rule, take $f(x) = 3x^2 \cdot 2x^3 = 6x^5$ , whose derivative is $30x^4$ . Applying the rule gives the same result:

\begin{aligned} (fg)' &= f'\,g + g'\,f \\ &= (6x)(2x^3) + (6x^2)(3x^2) \\ &= 12x^4 + 18x^4 = 30x^4. \end{aligned}

Summary

The derivative $f'(x_0)$ is the slope of $f$ at $x_0$ , defined as the limit of the difference quotient $\lim_{x \to x_0}\frac{f(x_0)-f(x)}{x_0 - x}$ .
A function is differentiable if this limit exists at every point of its domain; it fails at corners (like $|x|$ at $0$ ) and at any discontinuity.
The tangent line at $x_0$ is $y = f(x_0) + f'(x_0)(x - x_0)$ , and it is unique wherever the derivative exists.
Differentiability implies continuity, but not the reverse — continuity is necessary, not sufficient, for differentiability.
Core rules: the power rule $\frac{d}{dx}x^a = a x^{a-1}$ (all integers $a$ ), plus the sum, product, reciprocal, and quotient rules. The product rule is not the product of the derivatives.
Consequently, polynomials are differentiable everywhere and rational functions are differentiable wherever the denominator is nonzero.