Partial Derivatives

The partial derivative is the first tool of multivariable differential calculus. Its definition looks like “hold the other variables fixed and differentiate with respect to just one of them”, and what that operation means geometrically is the key to understanding the whole chapter.

Intuitively, a partial derivative is the steepest uphill slope of a surface in a particular direction. But watch out: the existence of the partial derivatives does not imply that the function is differentiable at that point — unlike the case of one variable, and this chapter gives a concrete counterexample. Whether the mixed partial derivatives are equal also has a precondition (usually that the second-order partial derivatives are continuous), and these preconditions are not formalism but necessary conditions for the conclusion to hold.

By the end of it you should be able to explain: why can a function still fail to be differentiable even though all its partial derivatives exist?