Documentation

Mathlib.Data.Polynomial.Derivative

The derivative map on polynomials #

Main definitions #

derivative p is the formal derivative of the polynomial p

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    theorem Polynomial.derivative_apply {R : Type u} [Semiring R] (p : Polynomial R) :
    Polynomial.derivative p = Polynomial.sum p fun (n : ) (a : R) => Polynomial.C (a * n) * Polynomial.X ^ (n - 1)
    theorem Polynomial.coeff_derivative {R : Type u} [Semiring R] (p : Polynomial R) (n : ) :
    Polynomial.coeff (Polynomial.derivative p) n = Polynomial.coeff p (n + 1) * (n + 1)
    theorem Polynomial.derivative_zero {R : Type u} [Semiring R] :
    Polynomial.derivative 0 = 0
    @[simp]
    theorem Polynomial.iterate_derivative_zero {R : Type u} [Semiring R] {k : } :
    (Polynomial.derivative)^[k] 0 = 0
    @[simp]
    theorem Polynomial.derivative_monomial {R : Type u} [Semiring R] (a : R) (n : ) :
    Polynomial.derivative ((Polynomial.monomial n) a) = (Polynomial.monomial (n - 1)) (a * n)
    theorem Polynomial.derivative_C_mul_X {R : Type u} [Semiring R] (a : R) :
    Polynomial.derivative (Polynomial.C a * Polynomial.X) = Polynomial.C a
    theorem Polynomial.derivative_C_mul_X_pow {R : Type u} [Semiring R] (a : R) (n : ) :
    Polynomial.derivative (Polynomial.C a * Polynomial.X ^ n) = Polynomial.C (a * n) * Polynomial.X ^ (n - 1)
    theorem Polynomial.derivative_C_mul_X_sq {R : Type u} [Semiring R] (a : R) :
    Polynomial.derivative (Polynomial.C a * Polynomial.X ^ 2) = Polynomial.C (a * 2) * Polynomial.X
    @[simp]
    theorem Polynomial.derivative_X_pow {R : Type u} [Semiring R] (n : ) :
    Polynomial.derivative (Polynomial.X ^ n) = Polynomial.C n * Polynomial.X ^ (n - 1)
    theorem Polynomial.derivative_X_sq {R : Type u} [Semiring R] :
    Polynomial.derivative (Polynomial.X ^ 2) = Polynomial.C 2 * Polynomial.X
    @[simp]
    theorem Polynomial.derivative_C {R : Type u} [Semiring R] {a : R} :
    Polynomial.derivative (Polynomial.C a) = 0
    theorem Polynomial.derivative_of_natDegree_zero {R : Type u} [Semiring R] {p : Polynomial R} (hp : Polynomial.natDegree p = 0) :
    Polynomial.derivative p = 0
    @[simp]
    theorem Polynomial.derivative_X {R : Type u} [Semiring R] :
    Polynomial.derivative Polynomial.X = 1
    @[simp]
    theorem Polynomial.derivative_one {R : Type u} [Semiring R] :
    Polynomial.derivative 1 = 0
    theorem Polynomial.derivative_bit0 {R : Type u} [Semiring R] {a : Polynomial R} :
    Polynomial.derivative (bit0 a) = bit0 (Polynomial.derivative a)
    theorem Polynomial.derivative_bit1 {R : Type u} [Semiring R] {a : Polynomial R} :
    Polynomial.derivative (bit1 a) = bit0 (Polynomial.derivative a)
    theorem Polynomial.derivative_add {R : Type u} [Semiring R] {f : Polynomial R} {g : Polynomial R} :
    Polynomial.derivative (f + g) = Polynomial.derivative f + Polynomial.derivative g
    theorem Polynomial.derivative_X_add_C {R : Type u} [Semiring R] (c : R) :
    Polynomial.derivative (Polynomial.X + Polynomial.C c) = 1
    theorem Polynomial.derivative_sum {R : Type u} {ι : Type y} [Semiring R] {s : Finset ι} {f : ιPolynomial R} :
    Polynomial.derivative (Finset.sum s fun (b : ι) => f b) = Finset.sum s fun (b : ι) => Polynomial.derivative (f b)
    theorem Polynomial.derivative_smul {R : Type u} [Semiring R] {S : Type u_1} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (s : S) (p : Polynomial R) :
    Polynomial.derivative (s p) = s Polynomial.derivative p
    @[simp]
    theorem Polynomial.iterate_derivative_smul {R : Type u} [Semiring R] {S : Type u_1} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (s : S) (p : Polynomial R) (k : ) :
    (Polynomial.derivative)^[k] (s p) = s (Polynomial.derivative)^[k] p
    @[simp]
    theorem Polynomial.iterate_derivative_C_mul {R : Type u} [Semiring R] (a : R) (p : Polynomial R) (k : ) :
    (Polynomial.derivative)^[k] (Polynomial.C a * p) = Polynomial.C a * (Polynomial.derivative)^[k] p
    theorem Polynomial.of_mem_support_derivative {R : Type u} [Semiring R] {p : Polynomial R} {n : } (h : n Polynomial.support (Polynomial.derivative p)) :
    theorem Polynomial.degree_derivative_lt {R : Type u} [Semiring R] {p : Polynomial R} (hp : p 0) :
    Polynomial.degree (Polynomial.derivative p) < Polynomial.degree p
    theorem Polynomial.natDegree_iterate_derivative {R : Type u} [Semiring R] (p : Polynomial R) (k : ) :
    Polynomial.natDegree ((Polynomial.derivative)^[k] p) Polynomial.natDegree p - k
    @[simp]
    theorem Polynomial.derivative_nat_cast {R : Type u} [Semiring R] {n : } :
    Polynomial.derivative n = 0
    @[simp]
    theorem Polynomial.derivative_ofNat {R : Type u} [Semiring R] (n : ) [Nat.AtLeastTwo n] :
    Polynomial.derivative (OfNat.ofNat n) = 0
    theorem Polynomial.iterate_derivative_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} {x : } (hx : Polynomial.natDegree p < x) :
    (Polynomial.derivative)^[x] p = 0
    @[simp]
    theorem Polynomial.iterate_derivative_C {R : Type u} {a : R} [Semiring R] {k : } (h : 0 < k) :
    (Polynomial.derivative)^[k] (Polynomial.C a) = 0
    @[simp]
    theorem Polynomial.iterate_derivative_one {R : Type u} [Semiring R] {k : } (h : 0 < k) :
    (Polynomial.derivative)^[k] 1 = 0
    @[simp]
    theorem Polynomial.iterate_derivative_X {R : Type u} [Semiring R] {k : } (h : 1 < k) :
    (Polynomial.derivative)^[k] Polynomial.X = 0
    theorem Polynomial.eq_C_of_derivative_eq_zero {R : Type u} [Semiring R] [NoZeroSMulDivisors R] {f : Polynomial R} (h : Polynomial.derivative f = 0) :
    f = Polynomial.C (Polynomial.coeff f 0)
    @[simp]
    theorem Polynomial.derivative_mul {R : Type u} [Semiring R] {f : Polynomial R} {g : Polynomial R} :
    Polynomial.derivative (f * g) = Polynomial.derivative f * g + f * Polynomial.derivative g
    theorem Polynomial.derivative_eval {R : Type u} [Semiring R] (p : Polynomial R) (x : R) :
    Polynomial.eval x (Polynomial.derivative p) = Polynomial.sum p fun (n : ) (a : R) => a * n * x ^ (n - 1)
    @[simp]
    theorem Polynomial.derivative_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (p : Polynomial R) (f : R →+* S) :
    Polynomial.derivative (Polynomial.map f p) = Polynomial.map f (Polynomial.derivative p)
    @[simp]
    theorem Polynomial.iterate_derivative_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (p : Polynomial R) (f : R →+* S) (k : ) :
    (Polynomial.derivative)^[k] (Polynomial.map f p) = Polynomial.map f ((Polynomial.derivative)^[k] p)
    theorem Polynomial.derivative_nat_cast_mul {R : Type u} [Semiring R] {n : } {f : Polynomial R} :
    Polynomial.derivative (n * f) = n * Polynomial.derivative f
    @[simp]
    theorem Polynomial.iterate_derivative_nat_cast_mul {R : Type u} [Semiring R] {n : } {k : } {f : Polynomial R} :
    (Polynomial.derivative)^[k] (n * f) = n * (Polynomial.derivative)^[k] f
    @[simp]
    theorem Polynomial.degree_derivative_eq {R : Type u} [Semiring R] [NoZeroSMulDivisors R] (p : Polynomial R) (hp : 0 < Polynomial.natDegree p) :
    Polynomial.degree (Polynomial.derivative p) = (Polynomial.natDegree p - 1)
    theorem Polynomial.coeff_iterate_derivative {R : Type u} [Semiring R] {k : } (p : Polynomial R) (m : ) :
    Polynomial.coeff ((Polynomial.derivative)^[k] p) m = Nat.descFactorial (m + k) k Polynomial.coeff p (m + k)
    theorem Polynomial.iterate_derivative_mul {R : Type u} [Semiring R] {n : } (p : Polynomial R) (q : Polynomial R) :
    (Polynomial.derivative)^[n] (p * q) = Finset.sum (Finset.range (Nat.succ n)) fun (k : ) => Nat.choose n k ((Polynomial.derivative)^[n - k] p * (Polynomial.derivative)^[k] q)
    theorem Polynomial.derivative_pow_succ {R : Type u} [CommSemiring R] (p : Polynomial R) (n : ) :
    Polynomial.derivative (p ^ (n + 1)) = Polynomial.C (n + 1) * p ^ n * Polynomial.derivative p
    theorem Polynomial.derivative_pow {R : Type u} [CommSemiring R] (p : Polynomial R) (n : ) :
    Polynomial.derivative (p ^ n) = Polynomial.C n * p ^ (n - 1) * Polynomial.derivative p
    theorem Polynomial.derivative_sq {R : Type u} [CommSemiring R] (p : Polynomial R) :
    Polynomial.derivative (p ^ 2) = Polynomial.C 2 * p * Polynomial.derivative p
    theorem Polynomial.pow_sub_one_dvd_derivative_of_pow_dvd {R : Type u} [CommSemiring R] {p : Polynomial R} {q : Polynomial R} {n : } (dvd : q ^ n p) :
    q ^ (n - 1) Polynomial.derivative p
    theorem Polynomial.pow_sub_dvd_iterate_derivative_of_pow_dvd {R : Type u} [CommSemiring R] {p : Polynomial R} {q : Polynomial R} {n : } (m : ) (dvd : q ^ n p) :
    q ^ (n - m) (Polynomial.derivative)^[m] p
    theorem Polynomial.pow_sub_dvd_iterate_derivative_pow {R : Type u} [CommSemiring R] (p : Polynomial R) (n : ) (m : ) :
    p ^ (n - m) (Polynomial.derivative)^[m] (p ^ n)
    theorem Polynomial.dvd_iterate_derivative_pow {R : Type u} [CommSemiring R] (f : Polynomial R) (n : ) {m : } (c : R) (hm : m 0) :
    n Polynomial.eval c ((Polynomial.derivative)^[m] (f ^ n))
    theorem Polynomial.iterate_derivative_X_pow_eq_nat_cast_mul {R : Type u} [CommSemiring R] (n : ) (k : ) :
    (Polynomial.derivative)^[k] (Polynomial.X ^ n) = (Nat.descFactorial n k) * Polynomial.X ^ (n - k)
    theorem Polynomial.iterate_derivative_X_pow_eq_C_mul {R : Type u} [CommSemiring R] (n : ) (k : ) :
    (Polynomial.derivative)^[k] (Polynomial.X ^ n) = Polynomial.C (Nat.descFactorial n k) * Polynomial.X ^ (n - k)
    theorem Polynomial.iterate_derivative_X_pow_eq_smul {R : Type u} [CommSemiring R] (n : ) (k : ) :
    (Polynomial.derivative)^[k] (Polynomial.X ^ n) = (Nat.descFactorial n k) Polynomial.X ^ (n - k)
    theorem Polynomial.derivative_X_add_C_pow {R : Type u} [CommSemiring R] (c : R) (m : ) :
    Polynomial.derivative ((Polynomial.X + Polynomial.C c) ^ m) = Polynomial.C m * (Polynomial.X + Polynomial.C c) ^ (m - 1)
    theorem Polynomial.derivative_X_add_C_sq {R : Type u} [CommSemiring R] (c : R) :
    Polynomial.derivative ((Polynomial.X + Polynomial.C c) ^ 2) = Polynomial.C 2 * (Polynomial.X + Polynomial.C c)
    theorem Polynomial.iterate_derivative_X_add_pow {R : Type u} [CommSemiring R] (n : ) (k : ) (c : R) :
    (Polynomial.derivative)^[k] ((Polynomial.X + Polynomial.C c) ^ n) = Nat.descFactorial n k (Polynomial.X + Polynomial.C c) ^ (n - k)
    theorem Polynomial.derivative_comp {R : Type u} [CommSemiring R] (p : Polynomial R) (q : Polynomial R) :
    Polynomial.derivative (Polynomial.comp p q) = Polynomial.derivative q * Polynomial.comp (Polynomial.derivative p) q
    theorem Polynomial.derivative_eval₂_C {R : Type u} [CommSemiring R] (p : Polynomial R) (q : Polynomial R) :
    Polynomial.derivative (Polynomial.eval₂ Polynomial.C q p) = Polynomial.eval₂ Polynomial.C q (Polynomial.derivative p) * Polynomial.derivative q

    Chain rule for formal derivative of polynomials.

    theorem Polynomial.derivative_prod {R : Type u} {ι : Type y} [CommSemiring R] [DecidableEq ι] {s : Multiset ι} {f : ιPolynomial R} :
    Polynomial.derivative (Multiset.prod (Multiset.map f s)) = Multiset.sum (Multiset.map (fun (i : ι) => Multiset.prod (Multiset.map f (Multiset.erase s i)) * Polynomial.derivative (f i)) s)
    theorem Polynomial.derivative_neg {R : Type u} [Ring R] (f : Polynomial R) :
    Polynomial.derivative (-f) = -Polynomial.derivative f
    @[simp]
    theorem Polynomial.iterate_derivative_neg {R : Type u} [Ring R] {f : Polynomial R} {k : } :
    (Polynomial.derivative)^[k] (-f) = -(Polynomial.derivative)^[k] f
    theorem Polynomial.derivative_sub {R : Type u} [Ring R] {f : Polynomial R} {g : Polynomial R} :
    Polynomial.derivative (f - g) = Polynomial.derivative f - Polynomial.derivative g
    theorem Polynomial.derivative_X_sub_C {R : Type u} [Ring R] (c : R) :
    Polynomial.derivative (Polynomial.X - Polynomial.C c) = 1
    @[simp]
    theorem Polynomial.iterate_derivative_sub {R : Type u} [Ring R] {k : } {f : Polynomial R} {g : Polynomial R} :
    (Polynomial.derivative)^[k] (f - g) = (Polynomial.derivative)^[k] f - (Polynomial.derivative)^[k] g
    @[simp]
    theorem Polynomial.derivative_int_cast {R : Type u} [Ring R] {n : } :
    Polynomial.derivative n = 0
    theorem Polynomial.derivative_int_cast_mul {R : Type u} [Ring R] {n : } {f : Polynomial R} :
    Polynomial.derivative (n * f) = n * Polynomial.derivative f
    @[simp]
    theorem Polynomial.iterate_derivative_int_cast_mul {R : Type u} [Ring R] {n : } {k : } {f : Polynomial R} :
    (Polynomial.derivative)^[k] (n * f) = n * (Polynomial.derivative)^[k] f
    theorem Polynomial.derivative_comp_one_sub_X {R : Type u} [CommRing R] (p : Polynomial R) :
    Polynomial.derivative (Polynomial.comp p (1 - Polynomial.X)) = -Polynomial.comp (Polynomial.derivative p) (1 - Polynomial.X)
    @[simp]
    theorem Polynomial.iterate_derivative_comp_one_sub_X {R : Type u} [CommRing R] (p : Polynomial R) (k : ) :
    (Polynomial.derivative)^[k] (Polynomial.comp p (1 - Polynomial.X)) = (-1) ^ k * Polynomial.comp ((Polynomial.derivative)^[k] p) (1 - Polynomial.X)
    theorem Polynomial.eval_multiset_prod_X_sub_C_derivative {R : Type u} [CommRing R] [DecidableEq R] {S : Multiset R} {r : R} (hr : r S) :
    Polynomial.eval r (Polynomial.derivative (Multiset.prod (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) S))) = Multiset.prod (Multiset.map (fun (a : R) => r - a) (Multiset.erase S r))
    theorem Polynomial.derivative_X_sub_C_pow {R : Type u} [CommRing R] (c : R) (m : ) :
    Polynomial.derivative ((Polynomial.X - Polynomial.C c) ^ m) = Polynomial.C m * (Polynomial.X - Polynomial.C c) ^ (m - 1)
    theorem Polynomial.derivative_X_sub_C_sq {R : Type u} [CommRing R] (c : R) :
    Polynomial.derivative ((Polynomial.X - Polynomial.C c) ^ 2) = Polynomial.C 2 * (Polynomial.X - Polynomial.C c)
    theorem Polynomial.iterate_derivative_X_sub_pow {R : Type u} [CommRing R] (n : ) (k : ) (c : R) :
    (Polynomial.derivative)^[k] ((Polynomial.X - Polynomial.C c) ^ n) = Nat.descFactorial n k (Polynomial.X - Polynomial.C c) ^ (n - k)
    theorem Polynomial.iterate_derivative_X_sub_pow_self {R : Type u} [CommRing R] (n : ) (c : R) :
    (Polynomial.derivative)^[n] ((Polynomial.X - Polynomial.C c) ^ n) = (Nat.factorial n)