The Derivative as a Limit

The derivative is not a new idea bolted onto limits. It is a limit — one specific limit that turns out to be useful everywhere.

From average to instantaneous

Between $x = a$ and $x = a + h$, the average rate of change of $f$ is

This is the difference quotient: rise over run, the slope of the line joining the two points on the graph (the secant line).

Now shrink $h$. The second point slides toward the first, the secant pivots, and in the limit you have the slope at a single point:

That is the definition. When the limit exists, $f$ is differentiable at $a$.

Why the limit is unavoidable

Setting $h = 0$ directly gives $\dfrac{f(a)-f(a)}{0} = \dfrac{0}{0}$ — indeterminate, exactly the form from the previous topic. A rate of change at a single instant is not something arithmetic can produce, because a single point has no run to divide by. The limit is what makes the question answerable.

Two readings of the same number

If $s(t)$ is position, $s'(t)$ is velocity and $s''(t)$ is acceleration. The second derivative is the rate of change of the rate of change — which is why acceleration feels different from speed.

Differentiable implies continuous — not the reverse

If $f$ is differentiable at $a$ then $f$ is continuous at $a$. The converse fails, and the standard counterexample is worth knowing:

It is continuous there. But

The one-sided limits of the difference quotient disagree, so $f'(0)$ does not exist. The graph has a corner: no single tangent line.

Continuity is necessary for differentiability, not sufficient.

The alternative form

Sometimes written with $x \to a$ instead of $h \to 0$:

Identical content, substituting $x = a + h$. Use whichever makes the algebra shorter.

Notation

All the same object. Leibniz's $dy/dx$ is suggestive of the difference quotient it came from, which is why it dominates in applied work.