> [!NOTE] Lemma (The derivative of $\log$) > Let $f(x)=\log x$, the [[Real Natural Logarithm Function|natural logarithm]], then its [[Fréchet Differentiation|derivative]] is given by $f'(x) = \frac{1}{x}$ Proof*. Using [[Derivative of Inverse of Strictly Monotonic Differentiable Real Function|derivative of inverses]], $f'(x) = \frac{1}{\exp ( \log(x))} = \frac{1}{x}$since the [[Derivative of Real Exponential Function|derivative]] of $x \mapsto \exp(x)$ is $x\mapsto \exp(x)$.