English
Under differentiability assumptions on c and d within s at x, the derivative within s of the product c d equals the product-derivative formula.
Русский
При предположениях дифференцируемости внутри s функций c и d в точке x, производная внутри s от произведения c d равна формуле произведения производных.
LaTeX
$$$\operatorname{derivWithin}(c \cdot d) \, s \, x = (\operatorname{derivWithin} c \, s \, x) \cdot d(x) + c(x) \cdot (\operatorname{derivWithin} d \, s \, x)$$$
Lean4
theorem derivWithin_fun_mul (hc : DifferentiableWithinAt 𝕜 c s x) (hd : DifferentiableWithinAt 𝕜 d s x) :
derivWithin (fun y => c y * d y) s x = derivWithin c s x * d x + c x * derivWithin d s x :=
by
by_cases hsx : UniqueDiffWithinAt 𝕜 s x
· exact (hc.hasDerivWithinAt.mul hd.hasDerivWithinAt).derivWithin hsx
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx]
