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 $\frac{dx}{d\phi}\big|_{\phi^+} = -r \sin\theta \sin\phi + \cos \phi \frac{d r \sin\theta}{d\phi}\big|_{\phi^+}$. From this post about the derivative of a constant function, $ \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^+}$. From this post about the derivative of a constant function, $ \boxed{ \frac{dz}{d\theta}\bigg|_{\theta^+} = -r \sin\theta }$. And with that, all nine derivatives of this intimidating matrix have been determined.
A Web Resource for Math