Science:Math Exam Resources/Courses/MATH200/December 2012/Question 03 (a)/Solution 2

Implicitly differentiating term by term, we find

$\begin{aligned} \frac{\partial x}{\partial x} (y z) + \frac{\partial z}{\partial x} (x y) + \frac{\partial x}{\partial x} + 3 z^{2} \frac{\partial z}{\partial x} & = 0 \\ \frac{\partial z}{\partial x} (x y + 3 z^{2}) & = - (y z + 1) \\ \frac{\partial z}{\partial x} & = - \frac{y z + 1}{x y + 3 z^{2}} \end{aligned}$