Theorem: d/d mu [-(x - mu)^2/(2 sigma^2)] at mu = x equals 0 (MLE first-order)

Theorem (MLE-Gaussian-first-order canonical): for the Gaussian log-likelihood -(x - mu)^2/(2 sigma^2), differentiation gives d/d mu = (x - mu)/sigma^2; substituting mu = x yields 0 identically. Canonical sympy pins: x_, mu, sigma =…