Theorems · Theorem · real analysis

Real.hasStrictFDerivAt_rpow_of_pos

∀ (p : ℝ × ℝ),
  0 < p.1 →
    HasStrictFDerivAt (fun x => x.1 ^ x.2)
      ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
        (p.1 ^ p.2 * Real.log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ)
      p

(x, y) ↦ x ^ y is strictly differentiable at p : ℝ × ℝ such that 0 < p.fst.

Defined in

Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Cited by

4 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound