deriv_log_comp_eq_logDeriv — Mathlib

English

For any n and x, ContDiffAt log x holds exactly when x ≠ 0.

Русский

Для любого n и точки x ContDiffAt log x происходит тогда и только тогда, когда x ≠ 0.

LaTeX

$$ContDiffAt ℝ n log x ↔ x ≠ 0$$

Lean4

/-- The derivative of `log ∘ f` is the logarithmic derivative provided `f` is differentiable and
`f x  ≠ 0`. -/
theorem deriv_log_comp_eq_logDeriv {f : ℝ → ℝ} {x : ℝ} (h₁ : DifferentiableAt ℝ f x) (h₂ : f x ≠ 0) :
    deriv (log ∘ f) x = logDeriv f x := by simp only [logDeriv, Pi.div_apply, ← deriv.log h₁ h₂, Function.comp_def]

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