Partial Derivatives

The partial derivative of with respect to at the point is if this limit exists.

In practice, we simply treat the other variables as constants when we take the partial derivative.

Example: If , then

title: Clairaut's Theorem
If $f_x, f _y$ and $f_{xy}$ exist near (a,b), and if $f_{xy}$ is continuous at (a,b), then $f_{yx}(a,b)$ also exists, and in fact $f_{yx}(a,b) = f_{xy}(a,b)$.