Derivations

One chain, three views. A derivation, a proof, and a memory walk are the same ordered steps — so each lives here once, read your way: Prove for rigor,Memorize for a loci walk with a test mode. Where a step has a picture, see it first.

This is the crossover made literal: the pillars are the units a chain is built from, the proofs are where a claim is shown to be unbreakable, and the mind castles are how it's kept. A derivation is all three at once. Pick one and toggle the view:

DerivationTaylor's theorem, from the Mean Value Theorem

Taylor's theorem, from the Mean Value Theorem

See it

f(x) = sin xPₙ — Taylor about a

degree n3center a0.00

worst error within ±2 of a 0.2426Lagrange bound 0.6667 = 2ⁿ⁺¹/(n+1)!

The gold polynomial hugs the curve near a and peels away past it — add a degree and the good region widens. The leftover between the two curves is exactly the remainder R_n; since every derivative of sine is at most 1, it can never exceed |x−a|ⁿ⁺¹/(n+1)! — the Lagrange bound shown above, made visible. That bound going to zero is why the series above stays faithful.

f (b) = k = 0 \sum n \frac{f ^{(k)} ( a )}{k !} (b - a)^{k} + \frac{f ^{(n + 1)} ( c )}{( n + 1 )!} (b - a)^{n + 1}

Taylor's theorem is the Mean Value Theorem, iterated. The MVT is the $n = 0$ case; Cauchy's extended MVT, applied $n + 1$ times to the error against the yardstick $(x - a)^{n + 1}$ , pins the remainder to a single $(n + 1)$ -th order term.

1 / 7

Reading asFollowing along — One stop at a time, the picture before the symbols — you meet the move as an idea first. Best for understanding.

Why read one chain three ways?

A derivation shows how a result is reached; a proof guarantees it; a memory walk keeps it. Understanding, verification, and retention are different acts — so the same ordered steps live here once and change lenses on demand:

Follow (step view) — meet the idea one place at a time, the picture before the algebra.
Prove (Outline) — see every step at once and check that each truly follows from the last.
Recall — hide the step and rebuild it from the place, until it's yours.

Stop 1 of 7 · a climb up a lookout tower

Ground floor

A rope (the secant) stretches from a to b; somewhere a leaning ladder (a tangent) has exactly its slope.

The Mean Value Theorem — the base case (

n = 0

). The secant slope from

a

b

is matched by some tangent inside.

f (b) = f (a) + f^{'} (c) (b - a)

Connects. The base case is the live MVT on Proofs; the bound

∣ x - a ∣^{n + 1} / (n + 1)!

is the precision behind the delta method, asymptotic expansions, and the characteristic-function proof of the CLT.

The shape underneath

Cells: the powers $(x - a)^{k}$ — one brick per degree, stacked
The move: a process $f \mapsto P_{n}$ ; the remainder is what the map can't reach — its forgotten higher-order detail
Lens: Representation — change of basis into powers
Units: Change · Certainty · The frame
Shown: drag the Mean Value Theorem

Roadmap — what this chain grows into

The handwritten session — these steps worked by hand from the Taylor's Theorem anthology — q-series, binomial, metric/topology, MVT, sin/cos/exp
Binomial theorem & the algebra of coefficients — $(x + y)^{n}$ , factoring, difference of squares/cubes — the algebra that becomes power-series arithmetic
Metric & topology axioms — distance, open sets, neighborhoods — the ε that limits are built from
Limits & continuity, analytic and topological — two definitions of one idea
The three derivative limits — central, forward, backward — and when half-steps help or hurt
The integral as refined partitions — Riemann → Lebesgue as the mesh shrinks
Taylor of sin, cos, exp, log — cycle the derivatives; expand in $x$ , $(x - a)$ , or $h$
$\int e^{- x^{2}}$ and the bridge to probability — the Gaussian integral the series opens onto
Radius of convergence & the q-series world — where $(x - a)$ is stressed past its limit; $∣ q ∣ < 1$ and the nome
Hensel's analogy: ℤ ↔ ℂ[X] — primes as linear factors; the Taylor expansion of a polynomial

Want a chain added? The ones you reach for most are the ones worth memorizing — research@backporch.studio.