fderiv_eq_smul_deriv — Mathlib

English

The Frechet derivative fderiv 𝕜 f x, viewed as a linear map 𝕜 → F, acts by multiplying its input by the scalar deriv f x; equivalently, (fderiv 𝕜 f x : 𝕜 → F) y = (deriv f x) · y.

Русский

Фрéше-дипазон производной действует умножением входа на скаляр deriv f x: ( fderiv 𝕜 f x : 𝕜 → F ) y = (deriv f x) · y.

LaTeX

$$$$ (fderiv 𝕜 f x : 𝕜 \\to F)(y) = (\\\\ deriv f x) \\\\cdot y $$$$

Lean4

@[simp]
theorem fderiv_eq_smul_deriv (y : 𝕜) : (fderiv 𝕜 f x : 𝕜 → F) y = y • deriv f x :=
  by
  rw [← fderiv_deriv, ← ContinuousLinearMap.map_smul]
  simp only [smul_eq_mul, mul_one]

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