derivWithin_comp_mul_left — Mathlib

English

An alternate specialization of the left-mulation derivative equivalence for smul and composition.

Русский

Альтернативная специализация эквивалентности производной слева для умножения и композиции.

LaTeX

$$$$Eq (HasDerivWithinAt f f' (c \cdot s) (c \cdot x)) (HasDerivWithinAt (fun x => f (c x)) (c f') s x)$$$$

Lean4

theorem derivWithin_comp_mul_left : derivWithin (f <| c * ·) s x = c • derivWithin f (c • s) (c * x) :=
  by
  simp only [← smul_eq_mul]
  rw [← derivWithin_const_smul', derivWithin, derivWithin, fderivWithin_comp_smul_eq_fderivWithin_smul, Pi.smul_def]

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