Log-derivative formula

The \(\log\)-derivative formula states that for a function composition involving logarithms:

\[\frac{d}{dx}\log(f(x)) = \frac{1}{f(x)} \frac{df(x)}{dx}.\]

If we have a function \(f(\mathbf{x})\) (\(\mathbf{x} \in \mathbb{R}^n\)) and we’re looking for the gradient of \(\log(f(\mathbf{x}))\), we can apply the \(\log\)-derivative formula in the multivariable setting:

\[\nabla_{\mathbf{x}} \log(f(\mathbf{x})) = \frac{1}{f(\mathbf{x})} \nabla_{\mathbf{x}} f(\mathbf{x})\]

This is the vector calculus equivalent of the scalar \(\log\)-derivative formula. The gradient \(\nabla_{\mathbf{x}}\) gives us a vector of partial derivatives with respect to each component of \(\mathbf{x}\).

For each component \(x_i\) of the vector \(\mathbf{x}\), we have:

\[\frac{\partial}{\partial x_i} \log(f(\mathbf{x})) = \frac{1}{f(\mathbf{x})} \frac{\partial f(\mathbf{x})}{\partial x_i}\]

So the full gradient is:

\[\begin{split}\nabla_{\mathbf{x}} \log(f(\mathbf{x})) = \frac{1}{f(\mathbf{x})} \begin{bmatrix} \frac{\partial f(\mathbf{x})}{\partial x_1} \\ \frac{\partial f(\mathbf{x})}{\partial x_2} \\ \vdots \\ \frac{\partial f(\mathbf{x})}{\partial x_n} \end{bmatrix}\end{split}\]

This generalizes the scalar \(\log\)-derivative to the vector setting, maintaining the same core principle: the gradient of the logarithm of a function equals the gradient of the function divided by the function itself.