Every derivative order of a quartic

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

f^{(0)}

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

(f^{(0)})'

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

(f')'

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

(f'')'

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

(f^{(3)})'

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

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

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

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

f''(2)=-36

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

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

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 orders

Each order uses differentiability of the preceding polynomial.

Zero and above-degree orders

Order 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

Same-source full JSON (formulas, ASTs, controls and revision log)