I would like to evaluate two more derivatives. They are $\frac{dx}{d\phi}\big|_{\phi^+}$ and $\frac{dz}{d\theta}\big|_{\theta^+}$ given $x = r \sin\theta\cos\phi$ and $z = r\cos\theta$.

Start with $\frac{dx}{d\phi}\big|_{\phi^+}$. The first step is to substitute $x$ with $r \sin\theta \cos\phi$.

$\frac{dx}{d\phi}\big|_{\phi^+} = \frac{d r \sin\theta \cos \phi}{d\phi}\big|_{\phi^+}$.

The next step is to apply the product rule to the right side.

$\frac{dx}{d\phi}\big|_{\phi^+} = r \sin\theta \frac{d \cos \phi}{d\phi}\big|_{\phi^+} + \cos \phi \frac{dr \sin\theta}{d\phi}\big|_{\phi^+}$.

Using this derivative of cosine, the previous equation becomes

$\frac{dx}{d\phi}\big|_{\phi^+} = -r \sin\theta \sin\phi + \cos \phi \frac{d r \sin\theta}{d\phi}\big|_{\phi^+}$.

$\boxed{ \frac{dx}{d\phi}\bigg|_{\phi^+} = -r \sin\theta \sin\phi }$.

Second, evaluate $\frac{dz}{d\theta}\big|_{\theta^+}$. The first step is to substitute $z$ with $r \cos\theta$.

$\frac{dz}{d\theta}\big|_{\theta^+} = \frac{dr\cos\theta }{d\theta}\big|_{\theta^+}$.

The next step is to apply the product rule to the right side.

$\frac{dz}{d\theta}\big|_{\theta^+} = r \frac{d \cos\theta}{d\theta}\big|_{\theta^+} + \cos\theta \frac{dr}{d\theta} \big|_{\theta^+}$.

Using this derivative of cosine, the previous equation becomes

$\frac{dz}{d\theta}\big|_{\theta^+} = -r \sin\theta + \cos\theta \frac{dr}{d\theta} \big|_{\theta^+}$.

$\boxed{ \frac{dz}{d\theta}\bigg|_{\theta^+} = -r \sin\theta }$.

And with that, all nine derivatives of this intimidating matrix have been determined. Gottfried Wilhelm Leibniz, one of the founders of Calculus, studied law at Leipzig University in Germany.