Using the methods in this post, I would like to evaluate

$\frac{dz}{d\phi}\bigg|_{\phi^+}$

with

$z=r\cos\theta$.

Substituting, the expression to evaluate is

$ \frac{d (r \cos\theta )}{d\phi} \bigg|_{\phi^+}$.

Since $ r \cos\theta $ does not depend on $\phi$, $ r \cos\theta $ is a constant function with respect to $\phi$. From this post, it follows that

$ \boxed { \frac{dz}{d\phi} \bigg|_{\phi^+} = 0 }$.

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