comp_const_sub — Mathlib

English

If g is differentiable within at and h differentiable, their composition has a derivative within s with chain rule adjustment.

Русский

Если g дифференцируема внутри и h дифференцируема, композиция имеет производную внутри s по правилу цепи.

LaTeX

$$$$\text{HasDerivWithinAt } (g\circ h) s x = (g'\circ h')(h x)\, s x.$$$$

Lean4

/-- Translation in the domain does not change the derivative. -/
theorem comp_const_sub (a x : 𝕜) (hf : HasDerivAt f f' (a - x)) : HasDerivAt (fun x ↦ f (a - x)) (-f') x := by
  simpa [Function.comp_def] using HasDerivAt.scomp (𝕜 := 𝕜) x hf <| hasDerivAt_id' x |>.const_sub a

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