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Calculus I · Unit 2A · exploration

Differentiability, Little-o Notation, and the Best Local Linear Model

Differentiability as Controlled Error

The ordinary definition

f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h

can be rearranged into

f(a+h)=f(a)+f′(a)h+r(h),f(a+h)=f(a)+f'(a)h+r(h),

where

r(h)h→0.\frac{r(h)}h\to0.

Analysts write r(h)=o(h)r(h)=o(h), read "little-o of hh." The notation means that r(h)r(h) becomes insignificant compared with hh. Thus

f(a+h)=f(a)+f′(a)h+o(h).f(a+h)=f(a)+f'(a)h+o(h).

For f(x)=x2f(x)=x^2 at a=3a=3,

f(3+h)=9+6h+h2.f(3+h)=9+6h+h^2.

The linear part is 9+6h9+6h, and the remainder is h2h^2. Since h2/h=h→0h^2/h=h\to0, the quadratic remainder is genuinely smaller than the first-order change.

For f(x)=x² at a=3, the exact change is 6h+h². The tangent model keeps 6h; the discarded remainder h² becomes small not only in absolute size, but also relative to h.
Read this graph as text

The nonlinear remainder shrinks faster than the input step. For f(x)=x 2 at a=3 , the exact change is 6h+h 2 . The tangent model keeps 6h ; the discarded remainder h 2 becomes small not only in absolute size, but also relative to h . Near h=0 , the gold curve falls toward zero much faster than the blue V-shaped graph. Dividing the remainder by the step gives h 2/|h|=|h| , which also approaches zero. That relative comparison, rather than the mere fact that both quantities are small, is what little- o notation records.

The visual uses labeled positions, solid and dashed line styles, and written descriptions so the nonlinear remainder shrinks faster than the input step does not depend on color.

Why it matters: This figure turns the symbolic statement r(h)=o(h) into a comparison of scales. Students often hear that the tangent error is "small" without learning what kind of smallness differentiability requires. The figure must make clear that the remainder is negligible relative to the first-order input displacement, not merely that it tends to zero.

Visual study

For f(x)=x² at a=3, the exact change is 6h+h². The tangent model keeps 6h; the discarded remainder h² becomes small not only in absolute size, but also relative to h.

Concept

Why this definition generalizes

In several variables, division by a vector makes no sense. The error formulation survives: a function is differentiable when it equals a constant plus a linear map plus an error small relative to the input displacement. The derivative becomes a matrix or linear transformation.

Exercise

For f(x)=x3f(x)=x^3 at a=2a=2, expand f(2+h)f(2+h) and identify the constant, linear, and remainder terms.

Exercise

Show that the remainder divided by hh tends to zero.

Exercise

Explain why f(x)=∣x∣f(x)=|x| at 00 cannot have one linear coefficient that makes the relative error vanish from both sides.

After the explanation

Use the section idea

Reading lens

Use these optional explorations to see the deeper analysis behind familiar derivative rules.

Mental model

Differentiability is a local linear approximation property with consequences beyond computation.

Decision

Return here after the core path is secure, and connect each abstraction to a concrete derivative example.

Common trap

Collecting formal language without linking it to the local linear model it describes.

Check yourself

Can you restate the advanced claim in ordinary language and test it on an example?

Interactive checkadvanced-little-o-01

For f(x)=x2f(x)=x^2 at a=3a=3, identify the remainder after the linear term.

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Show hint

Expand (3+h)2(3+h)^2.

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Source & rights

Original instruction with traceable references.

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