fderiv_smul — Mathlib

English

The derivative of the product c·f is fderiv 𝕜 (c) at x times f plus c x times fderiv 𝕜 f x with the right-smul adjustment.

Русский

Производная произведения c·f равна c(x)·fderiv f(x) + fderiv c(x) смещается вправо на (f(x)).

LaTeX

$$$$ \\text{fderiv } 𝕜 (c · f) x = c x · fderiv 𝕜 f x + (fderiv 𝕜 c x).smulRight (f x) $$$$

Lean4

theorem fderiv_smul (hc : DifferentiableAt 𝕜 c x) (hf : DifferentiableAt 𝕜 f x) :
    fderiv 𝕜 (c • f) x = c x • fderiv 𝕜 f x + (fderiv 𝕜 c x).smulRight (f x) :=
  (hc.hasFDerivAt.smul hf.hasFDerivAt).fderiv

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