2.2Differentiability in One Variable

Updated 22 Jun 2026Chapter (PDF)

We seek to view the derivative in a way that generalizes to multiple dimensions. From our studies of functions $f: \bR \to \bR$ and $f: \bR^2 \to \bR$ , we have the idea of a tangent line/plane approximation. That is, as we get closer and closer to the point of tangency, our tangent line (resp. plane) and curve (resp. surface) get closer and closer. We derive a more mathematically precise way to describe this.

Let $f: \bR \to \bR$ be a function, and say $a$ is the point at which we want to construct our tangent line $l(x) = mx + b$ . Our first obvious constraint on $l$ is that it must intersect $f$ at the point of tangency, meaning $l(a) = f(a)$ . This forces $ma + b = f(a)$ , and so $b = f(a) - ma$ . Plugging this back into our equation for the line, we get $l(x) = f(a) + m(x - a)$ . Thus, the family of all potential tangent lines reduces from all lines to only the lines passing through $(a, f(a))$ . Great! But how do we pin down the correct slope $m$ ?

[2.2.1]Example#

To see what goes wrong with a bad guess, let's isolate a specific example: $f(x) = x^2$ at $a = 0$ . Since $f(0) = 0$ , our candidate tangent lines take the form $l(x) = mx$ . So...is $l(x) = 5x$ a tangent line? Obviously not—we want our tangent line to fit snugly with $f$ near $x = 0$ .

Try slopes like $2x$ , $x$ , or $0.0001x$ . At some scale, these lines might look tangent to $f$ at $x = 0$ , but if we zoom in close enough, they all inevitably stab through the parabola rather than hugging it. While the absolute gap between the curve and the line vanishes as $x \to 0$ , it shrinks at the same rate that $x$ itself is shrinking.

Returning to our general $f(x)$ and $l(x)$ , we see that merely requiring $l(x) \to f(x)$ as $x \to a$ isn't enough; any intersecting line satisfies that! For $l(x)$ to fit snugly with $f(x)$ as the true tangent line, we want $l(x)$ to get closer to $f(x)$ at a rate faster than $x \to a$ . If the gap between $f(x)$ and $l(x)$ shrinks at the same rate as $x \to a$ , then the relative error between them remains constant, meaning the line intersects the curve at a sharp angle. To fit our notion of “tangency,” the error must be a smaller order of magnitude than the distance we are zooming in.

[2.2.2]Definition(Tangent Line)#

This behavior is perfectly captured by little- $o$ notation. We demand that the error $f(x) - l(x)$ is $o(|x - a|)$ . In other words, as $x \to a$ , the ratio of the error to the distance $|x - a|$ must completely vanish:

\lim_{x \to a} \frac{f(x) - l(x)}{|x - a|} = 0.

[2.2.3]Theorem(Equivalence of Little-

o

and Limit Definitions)#

Suppose $f: \bR \to \bR$ is a function, and let $l(x) = f(a) + m(x - a)$ be a generic line passing through $(a, f(a))$ . The error $f(x) - l(x)$ is $o(|x - a|)$ if and only if $m = f'(a)$ .

Proof.

By definition, the error being $o(|x - a|)$ implies that

\lim_{x \to a} \frac{f(x) - (f(a) + m(x - a))}{x - a} = 0.

Splitting the fraction, we get

\lim_{x \to a} \left( \frac{f(x) - f(a)}{x - a} - \frac{m(x - a)}{x - a} \right) = 0,

which evaluates to

\lim_{x \to a} \left( \frac{f(x) - f(a)}{x - a} \right) - m = 0.

Rearranging to solve for $m$ yields

m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a},

which is exactly $f'(a)$ . By simply enforcing that a generic line intersects $f(a)$ and that its error is $o(|x - a|)$ , the only valid slope remaining is exactly the standard limit definition of the derivative.