The Gradient of Log Loss on a Negative Example

A binary classifier turns a raw score $z$ into a probability with the sigmoid

$p(z) = \frac{1}{1 + e^{-z}}$

For a training example whose true label is $\mathbf{0}$, the correct answer has probability $1 - p$, so the cross-entropy loss is

$L(z) = -\ln\left(1 - p(z)\right)$

Backpropagation needs the gradient of the loss with respect to the raw score, all the way through both the logarithm and the sigmoid.

At the point $z = \ln 3$, what is $\dfrac{dL}{dz}$?