hasStrictDerivAt_norm_smul_pos — Mathlib

English

For t < 0, HasStrictFDerivAt of the norm at t x equals HasStrictFDerivAt of the negated function, i.e., scaling by a negative factor yields sign flip in derivative.

Русский

При t < 0 производная нормы при t x соответствует производной от −f, т. е. знак меняется на противоположный.

LaTeX

$$$t<0 \\Rightarrow HasStrictFDerivAt(\\|\\cdot\\|) f (t x) = HasStrictFDerivAt(\\|\\cdot\\|) (-f) (t x)$$$

Lean4

theorem hasStrictDerivAt_norm_smul_pos (ht : 0 < t) (h : HasStrictFDerivAt (‖·‖) f x) :
    HasStrictFDerivAt (‖·‖) f (t • x) := by simpa [ht] using h.hasStrictFDerivAt_norm_smul ht.ne'

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