Answer
$\frac{\partial z}{\partial x}=\frac{-x}{z-1}$
$\frac{\partial z}{\partial y}=\frac{y}{z-1}$
Work Step by Step
Use implicit differentiation to find the partial derivatives of the following function:
$x^2-y¨2+z^2-2z=4$
We first differentiate both sides of the equation with respect to $x$. All constants become zero:
$2x-0+2z\frac{\partial z}{\partial x}-2\frac{\partial z}{\partial x}=0$
$\frac{\partial z}{\partial x}=\frac{-2x}{2z-2}=\frac{-x}{z-1}$
We now differentiate both sides of the equation with respect to $y$, just as done above:
$0-2y+2z\frac{\partial z}{\partial y}-2\frac{\partial z}{\partial y}=0$
$\frac{\partial z}{\partial y}=\frac{2y}{2z-2}=\frac{y}{z-1}$
