Theorems · Theorem · real analysis

Real.hasStrictDerivAt_const_rpow_of_neg

∀ {a x : ℝ},
  a < 0 →
    HasStrictDerivAt (fun x => a ^ x)
      (a ^ x * Real.log a - Real.exp (Real.log a * x) * Real.sin (x * Real.pi) * Real.pi) x

This lemma says that fun x => a ^ x is strictly differentiable for a < 0. Note that these values of a are outside of the "official" domain of a ^ x, and we may redefine a ^ x for negative a if some other definition will be more convenient.

Defined in

Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Cited by

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Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound