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The derivative

The derivative at $𝑎$ is the slope of the tangent, written as a limit of slopes of secants. If $𝑓$ is defined near $𝑎$, I set

when that limit exists. The increment $h$ is allowed to approach from both sides; if the two one-sided limits disagree, there is no derivative.

The difference quotient is a function of $h$, for $h\neq 0$ small enough that $𝑎+h$ stays in the domain. Differentiability at $𝑎$ is a statement about that function as $h\to 0$, not a statement about $𝑓$ at points far from $𝑎$.

A computation that is actually a computation

For $𝑓(𝑥)={𝑥}^{2}$ the algebra is honest rather than mysterious:

Send $h\to 0$ and ${𝑓}^{\prime }(𝑎)=2𝑎$, which is (2.1) for this $𝑓$. The $h$ in the denominator cancelled against a genuine factor of $h$ in the numerator, not against a wish. The graph of that parabola is Figure 1.1.

The same pattern works for $𝑓(𝑥)={𝑥}^{3}$:

so ${𝑓}^{\prime }(𝑎)=3{𝑎}^{2}$. I am not yet claiming a general power rule; I am noticing that a polynomial identity supplies the factor of $h$ that the definition demands.

Linear approximation

Another way to say the same thing: $𝑓$ is differentiable at $𝑎$ when there is a number $𝑚$ such that

Then $𝑚$ is necessarily ${𝑓}^{\prime }(𝑎)$, and $𝑓(𝑎)+{𝑓}^{\prime }(𝑎)h$ is the best linear guess for $𝑓$ near $𝑎$. The error $𝑟(h)$ is smaller than linear. If I rewrite (2.2) this way, $𝑟(h)={h}^{2}$ for $𝑓(𝑥)={𝑥}^{2}$, and $\frac{𝑟(h)}{h}=h\to 0$.

A function that fails

Absolute value at $0$ is the standard refusal. The difference quotient is

The right-hand limit is $1$, the left-hand limit is $-1$. No single $𝑚$ works, so $𝑥\mapsto |𝑥|$ is not differentiable at $0$, even though it is continuous there. Continuity of $𝑓$ is necessary for ${𝑓}^{\prime }$ to exist, by Proposition 2.1, and it is not enough.