mul_const' — Mathlib

English

If ha and hb are differentiable within s at x, then (a b) has a derivative within s at x given by the product rule.

Русский

Если ha и hb дифференцируемы внутри s в x, то произведение a b имеет внутри s производную в x согласно правилу произведения.

LaTeX

$$$\\forall ha hb,\\text{HasFDerivWithinAt } (a \\cdot b) s x \ulikey \\text{has derivative by product rule}$$$

Lean4

@[fun_prop]
theorem mul_const' (ha : HasFDerivWithinAt a a' s x) (b : 𝔸) : HasFDerivWithinAt (fun y => a y * b) (a' <• b) s x :=
  ((ContinuousLinearMap.mul 𝕜 𝔸).flip b).hasFDerivAt.comp_hasFDerivWithinAt x ha

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