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The Derivative Is a Limit

We define the derivative as a limit of difference quotients, see what it means for the limit to exist, and prove that differentiability forces continuity.

From slope to limit

Pick a function f and a point a in its domain. For a nearby point x, the difference quotient (f(x) − f(a)) / (x − a) is the slope of the straight line — the secant — through the two graph points (a, f(a)) and (x, f(a)). It is an honest, computable number for every x ≠ a. The derivative is what this slope settles down to as x is squeezed toward a.

Formally, f is differentiable at a if the limit of the difference quotient as x → a exists. We call that number f′(a). The whole content of differentiation lives in this one limit; everything later is consequences. Writing h = x − a, the same definition reads as the limit of (f(a + h) − f(a)) / h as h → 0, which is often easier to compute with.

A derivative computed from the definition

Let us not assume any rules. We compute the derivative of f(x) = x² at an arbitrary point a directly from the limit, so you can see the machinery turning.

Claim: f(x) = x^2 is differentiable at every a, with f'(a) = 2a.

Difference quotient (h ≠ 0):
  (f(a+h) - f(a)) / h
    = ((a+h)^2 - a^2) / h
    = (a^2 + 2 a h + h^2 - a^2) / h
    = (2 a h + h^2) / h
    = 2a + h          (valid because h ≠ 0, so we may cancel)

Now take the limit as h -> 0:
  lim_{h->0} (2a + h) = 2a.

The limit exists, so f is differentiable at a and f'(a) = 2a.  ∎

Sanity check at a = 3:  f'(3) = 6, and the secant slopes 2(3)+h = 6+h
clearly approach 6 as h shrinks (h=0.1 -> 6.1, h=0.01 -> 6.01).
No shortcut rules — just the difference quotient, an algebraic simplification valid for h ≠ 0, and a limit.

Differentiable forces continuous

The first real theorem of the subject: if f is differentiable at a, then f is continuous at a. This is differentiability implies continuity. The proof is a one-line algebraic trick plus the algebra of limits.

Theorem: if f is differentiable at a, then f is continuous at a.

Proof. For x ≠ a write the identity
    f(x) - f(a) = ( (f(x) - f(a)) / (x - a) ) · (x - a).

As x -> a:
    first factor  -> f'(a)   (this is exactly the derivative)
    second factor -> 0        (since x - a -> 0)

By the algebra of limits, the product of the limits is the limit:
    lim_{x->a} ( f(x) - f(a) ) = f'(a) · 0 = 0,
hence lim_{x->a} f(x) = f(a).  That is continuity at a.  ∎
The derivative is finite, so multiplying it by something going to 0 gives 0.