Implicit Multivariable Differentiation

Implicit Function Theorem

1. $f(x,y,z)$, $f_z^{\prime }(x,y,z)$ are continuous near $(x_0,y_0,z_0)$
2. $f_z^{\prime }(x,y,z)\ne 0$
3. f is differentiable near $(x_0,y_0,z_0)⟸$ partials of $f$ are continuous

1, 2 $⟹$ There exists $z=z(x,y)$ such that $f(x,y,z(x,y))$ for all $(x,y)$ near $(x_0,y_0)$ and $z(x,y)$ is continuous

1,2,3 $⟹$ $z(x,y)$ is differentiable and

$$z_x^{\prime }=-\frac{f_x^{\prime }}{f_z^{\prime }},z_y^{\prime }=-\frac{f_y^{\prime }}{f_z^{\prime }}$$

Usage

$$z(x_0+\Delta x,y_0+\Delta y)=z(x_0,y_0)+z_x^{\prime }(x_0,y_0)\Delta x+z_y^{\prime }(x_0,y_0)\Delta y$$