clm_apply — Mathlib

English

If c has derivative c' at x and d has derivative d' at x, then the derivative of y ↦ (c(y) ∘ d(y)) at x is the sum of the appropriate left- and right-derivative contributions.

Русский

Если c имеет производную c' в x и d имеет производную d' в x, то производная f(y) = c(y) ∘ d(y) в x равна сумме левой и правой вкладов через c и d.

LaTeX

$$$ \operatorname{HasFDerivAt}\left( y \mapsto (c(y)) \circ (d(y)) \right) x = ( (c x) \circ d' ) + ( d' \text{ flipped} (c x) )$$$

Lean4

@[fun_prop]
theorem clm_apply (hc : Differentiable 𝕜 c) (hu : Differentiable 𝕜 u) : Differentiable 𝕜 fun y => (c y) (u y) :=
  fun x => (hc x).clm_apply (hu x)

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