According to my book the following function has a vanishing partial derivative with respect to $\phi$:
$$f(\phi(\theta), \phi^{'}(\theta), \theta)\ = \sqrt {1+sin^2(\theta)(\phi^{'})^2} $$
I can see that there is no explicit $\phi$-dependence, but I feel like I should be able to form the following expression using the chain rule:
$$\frac{\partial f}{\partial \phi}=\frac{\partial f}{\partial \phi^{'}}\frac{d \phi^{'}}{d \theta}\frac{d \theta}{d \phi}=\frac{\partial f}{\partial \phi^{'}}\frac{d^{2} \phi}{d \theta^{2}}\frac{1}{\phi^{'}}$$
It seems I have a fundamental misconception here, can anyone help me see what I'm missing?