# Every derivative order of a quartic

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This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-higher-order-derivative-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

Compute all derivative orders for low-to-high coefficients [2,-3,0,5,-2].

A full polynomial, without an analyticity claim for general smooth functions.





## Givens and symbols


- polynomial
- algebra
- definition_here


- G1 — f(x)=2-3x+5x^3-2x^4 on R, with derivative order k in N_0.


- f^{(k)} — k-th derivative
- c_j — low-to-high coefficients





## Rules and sources


- mko_definition · cited — The order n belongs to N_0 and every intermediate function must be differentiable again on the declared domain. Existence of the n-th derivative does not imply existence of the next. Exact polynomial families use falling-factorial coefficients and give zero derivatives above the polynomial degree. Finite-order or even all finite-order differentiability alone does not imply analyticity or Taylor-series convergence. · mko-higher-order-derivative
- polynomial · prerequisite — Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero.
- algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
- definition_here · prerequisite — The zeroth derivative is the original function; order k+1 differentiates the k-th derivative on the declared domain and requires that derivative to exist.





## Main derivation

S01 · definitionBefore

f^{(0)}

After

f^{(0)}(x)=f(x)=2-3x+5x^3-2x^4

Conditions: none

Rule: definition_here · Uses: G1

Leaf basis: definition_here

Order zero returns the original function; the coefficient array's quadratic term is zero.



S02 · impliesBefore

(f^{(0)})'

After

f'(x)=-3+15x^2-8x^3

Conditions: none

Rule: polynomial · Uses: G1, S01

Leaf basis: polynomial

Constants vanish, while each exponent multiplies its coefficient and decreases by one.



S03 · impliesBefore

(f')'

After

f''(x)=30x-24x^2

Conditions: none

Rule: polynomial · Uses: S02

Leaf basis: polynomial

Differentiate the preceding derivative, not the original function again.



S04 · impliesBefore

(f'')'

After

f^{(3)}(x)=30-48x

Conditions: none

Rule: polynomial · Uses: S03

Leaf basis: polynomial

The second derivative is polynomial and differentiable again on R.



S05 · impliesBefore

(f^{(3)})'

After

f^{(4)}(x)=-48

Conditions: none

Rule: polynomial · Uses: S04

Leaf basis: polynomial

The fourth derivative is constant, retaining the negative sign.



S06 · impliesBefore

(f^{(4)})'=0

After

f^{(k)}=0\quad\forall k\ge5

Conditions: none

Rule: polynomial · Uses: S05

Leaf basis: polynomial

The fifth derivative is zero; differentiating zero stays zero, so induction covers all higher orders.



S07 · equivalentBefore

f''(2)=30\cdot2-24\cdot2^2

After

f''(2)=-36

Conditions: none

Rule: algebra · Uses: S03

Leaf basis: algebra

Evaluate only after fixing the order and point.



S08 · impliesBefore

f(x)=\sum_{j=0}^{4}c_jx^j

After

f^{(k)}(x)=\sum_{j=k}^{4}c_j\frac{j!}{(j-k)!}x^{j-k}\ (0\le k\le4)

Conditions: 0\le k\le4

Rule: polynomial · Uses: G1, S02, S06

Leaf basis: polynomial

After k differentiations the coefficient is the falling factorial; orders above four were separately proved zero.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S02, S06 — The formula uses j≥k and never a negative factorial.
- S07 · inverse_equivalence · S03 — The square applies to 2, not to coefficient 24.
- S06 · premise_audit · S05 — This uses the full polynomial, not a general inference from existence of one derivative to the next.
- S05 · premise_audit · S04 — The preceding derivative exists throughout the domain before differentiating it.
- S04 · premise_audit · S03 — Coefficient 24 is multiplied by exponent 2, giving 48.
- S03 · premise_audit · S02 — The derivative of -3 is zero.
- S02 · premise_audit · G1, S01 — Every intermediate result remains polynomial, qualifying the next derivative.
- S01 · premise_audit · G1 — Order zero must not be mistaken for the zero function.





## Error controls and negative controls

Intermediate ordersEach order uses differentiability of the preceding polynomial.

Zero and above-degree ordersOrder zero returns f; only above-degree orders vanish.


- The zero-order derivative is zero. → reject — It is defined as f itself.
- f''(2)=60-48. → reject — The second term is -24*2^2=-96.
- The fifth derivative of this f is constant one. → reject — The fourth derivative is constant -48, whose derivative is zero; every higher derivative is zero too.





## Result and limitations

Orders one through four are as listed, all higher orders vanish, and f''(2)=-36.







## Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 615b09a85c29b8a1f99a598929f470b21db66cd8637938659f3f62523c06ab25
- original source SHA-256: 615b09a85c29b8a1f99a598929f470b21db66cd8637938659f3f62523c06ab25
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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