# mathlibdocumentation

analysis.calculus.iterated_deriv

# One-dimensional iterated derivatives #

We define the n-th derivative of a function f : 𝕜 → F as a function iterated_deriv n f : 𝕜 → F, as well as a version on domains iterated_deriv_within n f s : 𝕜 → F, and prove their basic properties.

## Main definitions and results #

Let 𝕜 be a nondiscrete normed field, and F a normed vector space over 𝕜. Let f : 𝕜 → F.

• iterated_deriv n f is the n-th derivative of f, seen as a function from 𝕜 to F. It is defined as the n-th Fréchet derivative (which is a multilinear map) applied to the vector (1, ..., 1), to take advantage of all the existing framework, but we show that it coincides with the naive iterative definition.
• iterated_deriv_eq_iterate states that the n-th derivative of f is obtained by starting from f and differentiating it n times.
• iterated_deriv_within n f s is the n-th derivative of f within the domain s. It only behaves well when s has the unique derivative property.
• iterated_deriv_within_eq_iterate states that the n-th derivative of f in the domain s is obtained by starting from f and differentiating it n times within s. This only holds when s has the unique derivative property.

## Implementation details #

The results are deduced from the corresponding results for the more general (multilinear) iterated Fréchet derivative. For this, we write iterated_deriv n f as the composition of iterated_fderiv 𝕜 n f and a continuous linear equiv. As continuous linear equivs respect differentiability and commute with differentiation, this makes it possible to prove readily that the derivative of the n-th derivative is the n+1-th derivative in iterated_deriv_within_succ, by translating the corresponding result iterated_fderiv_within_succ_apply_left for the iterated Fréchet derivative.

def iterated_deriv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] (n : ) (f : 𝕜 → F) (x : 𝕜) :
F

The n-th iterated derivative of a function from 𝕜 to F, as a function from 𝕜 to F.

Equations
• x = n f x) (λ (i : fin n), 1)
def iterated_deriv_within {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] (n : ) (f : 𝕜 → F) (s : set 𝕜) (x : 𝕜) :
F

The n-th iterated derivative of a function from 𝕜 to F within a set s, as a function from 𝕜 to F.

Equations
• x = f s x) (λ (i : fin n), 1)
theorem iterated_deriv_within_univ {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} :

### Properties of the iterated derivative within a set #

theorem iterated_deriv_within_eq_iterated_fderiv_within {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {s : set 𝕜} {x : 𝕜} :
x = f s x) (λ (i : fin n), 1)
theorem iterated_deriv_within_eq_equiv_comp {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {s : set 𝕜} :
= F).symm) s

Write the iterated derivative as the composition of a continuous linear equiv and the iterated Fréchet derivative

theorem iterated_fderiv_within_eq_equiv_comp {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {s : set 𝕜} :
s =

Write the iterated Fréchet derivative as the composition of a continuous linear equiv and the iterated derivative.

theorem iterated_fderiv_within_apply_eq_iterated_deriv_within_mul_prod {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {s : set 𝕜} {x : 𝕜} {m : fin n → 𝕜} :
f s x) m = (∏ (i : fin n), m i) x

The n-th Fréchet derivative applied to a vector (m 0, ..., m (n-1)) is the derivative multiplied by the product of the m is.

@[simp]
theorem iterated_deriv_within_zero {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {s : set 𝕜} :
= f
@[simp]
theorem iterated_deriv_within_one {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {s : set 𝕜} (hs : s) {x : 𝕜} (hx : x s) :
x = s x
theorem times_cont_diff_on_of_continuous_on_differentiable_on_deriv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {s : set 𝕜} {n : with_top } (Hcont : ∀ (m : ), m ncontinuous_on (λ (x : 𝕜), x) s) (Hdiff : ∀ (m : ), m < n (λ (x : 𝕜), x) s) :
f s

If the first n derivatives within a set of a function are continuous, and its first n-1 derivatives are differentiable, then the function is C^n. This is not an equivalence in general, but this is an equivalence when the set has unique derivatives, see times_cont_diff_on_iff_continuous_on_differentiable_on_deriv.

theorem times_cont_diff_on_of_differentiable_on_deriv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {s : set 𝕜} {n : with_top } (h : ∀ (m : ), m n s) s) :
f s

To check that a function is n times continuously differentiable, it suffices to check that its first n derivatives are differentiable. This is slightly too strong as the condition we require on the n-th derivative is differentiability instead of continuity, but it has the advantage of avoiding the discussion of continuity in the proof (and for n = ∞ this is optimal).

theorem times_cont_diff_on.continuous_on_iterated_deriv_within {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {s : set 𝕜} {n : with_top } {m : } (h : f s) (hmn : m n) (hs : s) :
s

On a set with unique derivatives, a C^n function has derivatives up to n which are continuous.

theorem times_cont_diff_on.differentiable_on_iterated_deriv_within {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {s : set 𝕜} {n : with_top } {m : } (h : f s) (hmn : m < n) (hs : s) :
s) s

On a set with unique derivatives, a C^n function has derivatives less than n which are differentiable.

theorem times_cont_diff_on_iff_continuous_on_differentiable_on_deriv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {s : set 𝕜} {n : with_top } (hs : s) :
f s (∀ (m : ), m n s) ∀ (m : ), m < n s) s

The property of being C^n, initially defined in terms of the Fréchet derivative, can be reformulated in terms of the one-dimensional derivative on sets with unique derivatives.

theorem iterated_deriv_within_succ {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {s : set 𝕜} {x : 𝕜} (hxs : x) :
iterated_deriv_within (n + 1) f s x = s x

The n+1-th iterated derivative within a set with unique derivatives can be obtained by differentiating the n-th iterated derivative.

theorem iterated_deriv_within_eq_iterate {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {s : set 𝕜} {x : 𝕜} (hs : s) (hx : x s) :
x = (λ (g : 𝕜 → F), s)^[n] f x

The n-th iterated derivative within a set with unique derivatives can be obtained by iterating n times the differentiation operation.

theorem iterated_deriv_within_succ' {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {s : set 𝕜} {x : 𝕜} (hxs : s) (hx : x s) :
iterated_deriv_within (n + 1) f s x = s x

The n+1-th iterated derivative within a set with unique derivatives can be obtained by taking the n-th derivative of the derivative.

### Properties of the iterated derivative on the whole space #

theorem iterated_deriv_eq_iterated_fderiv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {x : 𝕜} :
x = n f x) (λ (i : fin n), 1)
theorem iterated_deriv_eq_equiv_comp {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} :
= F).symm) f

Write the iterated derivative as the composition of a continuous linear equiv and the iterated Fréchet derivative

theorem iterated_fderiv_eq_equiv_comp {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} :
f =

Write the iterated Fréchet derivative as the composition of a continuous linear equiv and the iterated derivative.

theorem iterated_fderiv_apply_eq_iterated_deriv_mul_prod {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} {x : 𝕜} {m : fin n → 𝕜} :
n f x) m = (∏ (i : fin n), m i) x

The n-th Fréchet derivative applied to a vector (m 0, ..., m (n-1)) is the derivative multiplied by the product of the m is.

@[simp]
theorem iterated_deriv_zero {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} :
= f
@[simp]
theorem iterated_deriv_one {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} :
=
theorem times_cont_diff_iff_iterated_deriv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {n : with_top } :
f (∀ (m : ), m ncontinuous f)) ∀ (m : ), m < n f)

The property of being C^n, initially defined in terms of the Fréchet derivative, can be reformulated in terms of the one-dimensional derivative.

theorem times_cont_diff_of_differentiable_iterated_deriv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {n : with_top } (h : ∀ (m : ), m n f)) :
f

To check that a function is n times continuously differentiable, it suffices to check that its first n derivatives are differentiable. This is slightly too strong as the condition we require on the n-th derivative is differentiability instead of continuity, but it has the advantage of avoiding the discussion of continuity in the proof (and for n = ∞ this is optimal).

theorem times_cont_diff.continuous_iterated_deriv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {n : with_top } (m : ) (h : f) (hmn : m n) :
theorem times_cont_diff.differentiable_iterated_deriv {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {f : 𝕜 → F} {n : with_top } (m : ) (h : f) (hmn : m < n) :
f)
theorem iterated_deriv_succ {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} :
iterated_deriv (n + 1) f = deriv f)

The n+1-th iterated derivative can be obtained by differentiating the n-th iterated derivative.

theorem iterated_deriv_eq_iterate {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} :

The n-th iterated derivative can be obtained by iterating n times the differentiation operation.

theorem iterated_deriv_succ' {𝕜 : Type u_1} {F : Type u_2} [normed_group F] [ F] {n : } {f : 𝕜 → F} :
iterated_deriv (n + 1) f = (deriv f)

The n+1-th iterated derivative can be obtained by taking the n-th derivative of the derivative.