English
If hy: y = f x and hl differentiableWithinAt l t y, hf differentiableWithinAt f s x, hs maps f s t, then derivWithin of l ∘ f equals fderivWithin(l) applied to derivWithin f.
Русский
Если hy: y = f x и hl дифференцируемо внутри l, hf дифференцируемо внутри f, then derivWithin(l∘f) равна fderivWithin(l) применённой к derivWithin(f).
LaTeX
$$$$fderivWithin\ 𝕜 l t (f x) \land differentiableWithinAt 𝕜 f s x \land MapsTo f s t \Rightarrow derivWithin (l \circ f) s x = (fderivWithin 𝕜 l t (f x) : F \to E) (derivWithin f s x)$$$$
Lean4
theorem fderiv_comp_deriv (hl : DifferentiableAt 𝕜 l (f x)) (hf : DifferentiableAt 𝕜 f x) :
deriv (l ∘ f) x = (fderiv 𝕜 l (f x) : F → E) (deriv f x) :=
(hl.hasFDerivAt.comp_hasDerivAt x hf.hasDerivAt).deriv
