# Concept derivation guides 299 concept teaching guides, separate from the 36,000 individual question derivations. [Search all questions & derivations →](/en/explore/) [← Previous](/en/derivations/?page=10#results)[1](/en/derivations/?page=1#results) … [10](/en/derivations/?page=10#results) 11 [12](/en/derivations/?page=12#results) [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=12#results) 299 results · Page 11 of 13 Concept guide ## [Every derivative order of a quartic](/en/derivations/derivation-mko-higher-order-derivative-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Compute all derivative orders for low-to-high coefficients [2,-3,0,5,-2]. Higher-order derivative derivation-mko-higher-order-derivative-01 Concept guide ## [Separating every pair in a discrete three-point space](/en/derivations/derivation-mko-hausdorff-space-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Exhaustively audit Hausdorffness of a discrete three-point space and contrast a failed pair. Hausdorff space derivation-mko-hausdorff-space-01 Concept guide ## [A continuous CDF with a flat interval](/en/derivations/derivation-mko-continuous-real-distribution-function-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Construct a complete knot-based CDF and audit atomlessness. Continuous real distribution function derivation-mko-continuous-real-distribution-function-01 Concept guide ## [Catalan convolution and a divisibility certificate](/en/derivations/derivation-mko-catalan-numbers-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Generate C4 from C0 and check both closed forms. Catalan numbers derivation-mko-catalan-numbers-01 Concept guide ## [The complete nonzero square table modulo 7](/en/derivations/derivation-mko-quadratic-residue-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Modulo 7, the nonzero quadratic residues are {1,2,4}; the class of -5 equals the class of 2 and has root classes 3 and 4, while 3 has no square root modulo 7. Quadratic residue derivation-mko-quadratic-residue-01 Concept guide ## [An existential finite accepting NFA path](/en/derivations/derivation-mko-nondeterministic-finite-automaton-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) The specified NFA accepts aab despite a dead alternative branch; an epsilon loop alone does not accept. Nondeterministic finite automaton derivation-mko-nondeterministic-finite-automaton-01 Concept guide ## [Exact-order witnesses for the prime divisors](/en/derivations/derivation-mko-cauchy-theorem-finite-groups-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) In this six-element group, prime divisors 2,3 have exact-order witnesses s,r; there is no element of order 6. Cauchy theorem for finite groups derivation-mko-cauchy-theorem-finite-groups-01 Concept guide ## [A Taylor polynomial retaining the center and factorials](/en/derivations/derivation-mko-taylor-polynomial-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Construct T_2 at center a=1 for the specified cubic polynomial. Taylor polynomial derivation-mko-taylor-polynomial-01 Concept guide ## [Compactness reduces separating neighborhoods to finitely many](/en/derivations/derivation-mko-compact-subset-closed-in-hausdorff-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Prove a compact subset K of a Hausdorff space is closed. A compact subset of a Hausdorff space is closed derivation-mko-compact-subset-closed-in-hausdorff-01 Concept guide ## [Separate limits at every infinite endpoint and singularity](/en/derivations/derivation-mko-improper-integral-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Compute three explicit power-law improper integrals and distinguish principal value. Improper integral derivation-mko-improper-integral-01 Concept guide ## [Radius, endpoints and complete coefficient laws](/en/derivations/derivation-mko-power-series-radius-of-convergence-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Find R for geometric coefficients 2^n, and exhibit endpoint and zero/infinity boundaries. Radius of convergence of a power series derivation-mko-power-series-radius-of-convergence-01 Concept guide ## [The three-valued Legendre symbol from squares](/en/derivations/derivation-mko-legendre-symbol-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Modulo 7 the symbol on 0,1,2,3,4,5,6 is respectively 0,1,1,-1,1,-1,-1. Legendre symbol derivation-mko-legendre-symbol-01 Concept guide ## [The complete reachable-subset DFA of an epsilon-NFA](/en/derivations/derivation-mko-subset-construction-nfa-to-dfa-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) List every transition and prove language equivalence by prefix induction. Subset construction from an NFA to a DFA derivation-mko-subset-construction-nfa-to-dfa-01 Concept guide ## [Normality of rotations from every conjugate](/en/derivations/derivation-mko-finite-normal-subgroup-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) N={e,r,r2} is normal; H={e,s} is a subgroup but is not normal. Finite normal subgroup derivation-mko-finite-normal-subgroup-01 Concept guide ## [Addition and strict positivity of the full exponential series](/en/derivations/derivation-mko-real-exponential-function-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) From E(x)=sum x^n/n!, prove absolute convergence for all real x, addition, inverses and positivity. Real exponential function derivation-mko-real-exponential-function-01 Concept guide ## [A Fibonacci-type OGF with initial boundary terms](/en/derivations/derivation-mko-analytic-ordinary-generating-function-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Derive the OGF from a recurrence on a justified convergence domain. Analytic ordinary generating function derivation-mko-analytic-ordinary-generating-function-01 Concept guide ## [Two independent routes for Euler's criterion](/en/derivations/derivation-mko-euler-criterion-quadratic-residue-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) For p=7,a=-4 the Legendre value is -1 and a^3=6=-1 mod 7; 2 gives +1. Euler criterion for quadratic residues derivation-mko-euler-criterion-quadratic-residue-01 Concept guide ## [A scan decider for an even number of ones](/en/derivations/derivation-mko-turing-machine-decision-model-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Give the complete machine, trace, ranking and all-input correctness. Turing-machine decision model derivation-mko-turing-machine-decision-model-01 Concept guide ## [Coordinate gradients, stationarity and differentiability](/en/derivations/derivation-mko-gradient-vector-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Compute the gradient of a specified bivariate polynomial and audit two false inferences. Gradient vector derivation-mko-gradient-vector-01 Concept guide ## [Checking every edge and internal vertex of a feasible flow](/en/derivations/derivation-mko-finite-flow-network-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Audit a complete flow network with antiparallel edges, a zero-capacity edge and a zero-flow vertex. Finite flow network derivation-mko-finite-flow-network-01 Concept guide ## [Kernel, fibers and injectivity from C4 to C2](/en/derivations/derivation-mko-finite-group-homomorphism-kernel-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective. Kernel of a finite group homomorphism derivation-mko-finite-group-homomorphism-kernel-01 Concept guide ## [A quotient table checked on every representative pair](/en/derivations/derivation-mko-finite-quotient-group-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) G/N has cosets A=N,B={s,rs,r2s} and well-defined products AA=A,AB=BA=B,BB=A. Finite quotient group derivation-mko-finite-quotient-group-01 Concept guide ## [Full conjugation action and three classes](/en/derivations/derivation-mko-finite-group-conjugacy-class-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate. Conjugacy class in a finite group derivation-mko-finite-group-conjugacy-class-01 Concept guide ## [Standard-normal symmetry with explicit named facts](/en/derivations/derivation-mko-standard-normal-distribution-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D11) Separate Gaussian prerequisites from derived CDF symmetry. Standard normal distribution derivation-mko-standard-normal-distribution-01 [← Previous](/en/derivations/?page=10#results)[1](/en/derivations/?page=1#results) … [10](/en/derivations/?page=10#results) 11 [12](/en/derivations/?page=12#results) [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=12#results) 299 results · Page 11 of 13