Theorems · Theorem · real analysis

HasDerivAt.comp

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] (x : 𝕜) {𝕜' : Type u_1} [inst_1 : NontriviallyNormedField 𝕜']
  [inst_2 : NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' : 𝕜'},
  HasDerivAt h₂ h₂' (h x) → HasDerivAt h h' x → HasDerivAt (h₂ ∘ h) (h₂' * h') x

The chain rule. Note that the function h₂ is a function on an algebra. If you are looking for the chain rule with h₂ taking values in a vector space, use HasDerivAt.scomp.

Defined in

Mathlib.Analysis.Calculus.Deriv.Comp

Cited by

43 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAlgebra