Chapter 13.7, Problem 35ES

Prove: If the surfaces $z=f\left(x,y\right)\text{and}z=g\left(x,y\right)$ intersect at $P\left(x_0,y_0,z_0\right),$ and if $f\text{and}g$ are differentiable at $\left(x_0,y_0\right),$ then the normal lines at $P$ are perpendicular if and only if

$f_x\left(x_0,y_0\right)g_x\left(x_0,y_0\right)+f_y\left(x_0,y_0\right)g_y\left(x_0,y_0\right)=-1$