Theorems · Theorem · real analysis

Real.hasStrictFDerivAt_rpow_of_neg

∀ (p : ℝ × ℝ),
  p.1 < 0 →
    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 - Real.exp (Real.log p.1 * p.2) * Real.sin (p.2 * Real.pi) * Real.pi) •
          ContinuousLinearMap.snd ℝ ℝ ℝ)
      p

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

Defined in

Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Cited by

2 results in Mathlib

Foundations

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