What is derivative?

Example 14.2

The converse of “differentiable $\Rightarrow$ continuous” fails: $\abs{\,\cdot\,}$ at $0$. More surprisingly, differentiable does not imply $C^1$: the function $f(x) = x^2 \sin\frac 1x$ ($f(0) = 0$) is differentiable everywhere, with $f'(0) = 0$, but $f'(x) = 2x \sin\frac1x - \cos\frac 1x$ has no limit at $0$ (Exercise 14.2).

Example 14.3 (Differentiable at exactly one point)

Let $f(x) = x^2$ for $x \in \Q$ and $f(x) = 0$ for $x \notin \Q$. At $0$: $\bigl|\frac{f(h) - 0}{h}\bigr| \leq \abs h \to 0$, so $f$ is differentiable at $0$ with $f'(0) = 0$. At any $x_0 \neq 0$, $f$ is not even continuous: rational and irrational sequences converging to $x_0$ send $f$ to $x_0^2 \neq 0$ and to $0$ respectively (density, Theorem 10.14). So differentiability is a genuinely pointwise notion: it can hold at one point of $\R$ and nowhere else. The moral for practice: statements like the monotonicity criterion or Rolle require the derivative on an interval — possessing $f'(x_0)$ at isolated points, however many, supports no global conclusion whatsoever.

Example 14.6 (Inverse derivatives, twice)

The theorem recomputes the classical derivatives with no limit work. For $\ln = \exp^{-1}$: at $y = \eu^x$,

valid for every $y > 0$ since $\exp' = \exp$ never vanishes. For $\arctan = \tan^{-1}$: at $y = \tan x$,

using $\tan' = 1 + \tan^2 > 0$. The closing insight: the formula converts knowledge about a function into knowledge about its inverse at the price of one substitution — and the substitution ($x = \ln y$, $x = \arctan y$) is exactly the statement that the two variables live on opposite sides of the bijection.