const_cpow — Mathlib

English

Let f be differentiable within s at x and f(x) ∈ slitPlane. Then x ↦ c^{f(x)} has derivative c f(x)^{c-1} f'(x) within s at x.

Русский

Пусть f дифференцируема внутри s в x и f(x) ∈ slitPlane. Тогда x ↦ c^{f(x)} имеет производную внутри s: c f(x)^{c-1} f'(x).

LaTeX

$$$\\text{If }f\\text{ is differentiable within }s\\text{ at }x\\text{ and }f(x)\\in\\mathrm{slitPlane},\\ \\dfrac{d}{dx} c^{f(x)}= c f(x)^{c-1} f'(x).$$$

Lean4

theorem const_cpow (hf : HasDerivWithinAt f f' s x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
    HasDerivWithinAt (fun x => c ^ f x) (c ^ f x * Complex.log c * f') s x :=
  (hasStrictDerivAt_const_cpow h0).hasDerivAt.comp_hasDerivWithinAt x hf

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