Derivative of x ↦ f (cx)
#
In this file we prove that the derivative of fun x ↦ f (c * x)
equals c
times the derivative of f
evaluated at c * x
.
Since Mathlib uses 0
as the fallback value for the derivatives whenever they are undefined,
the theorems in this file require neither differentiability of f
,
nor assumptions like UniqueDiffWithinAt 𝕜 s x
.
theorem
hasDerivWithinAt_comp_mul_left_smul_iff
{𝕜 : Type u_1}
{E : Type u_2}
[NontriviallyNormedField 𝕜]
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{c : 𝕜}
{f : 𝕜 → E}
{f' : E}
{s : Set 𝕜}
{x : 𝕜}
:
theorem
derivWithin_comp_mul_left
{𝕜 : Type u_1}
{E : Type u_2}
[NontriviallyNormedField 𝕜]
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
(c : 𝕜)
(f : 𝕜 → E)
(s : Set 𝕜)
(x : 𝕜)
:
theorem
deriv_comp_mul_left
{𝕜 : Type u_1}
{E : Type u_2}
[NontriviallyNormedField 𝕜]
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
(c : 𝕜)
(f : 𝕜 → E)
(x : 𝕜)
: