differentiability in several variables
In one variable, 'differentiable' just means the graph has a tangent line. In several variables the honest meaning is: the graph has a tangent plane (or tangent flat) that hugs the surface so closely that the gap between them vanishes faster than you walk away from the point. This is a real promise of good local behavior, much stronger than the bookkeeping fact that a few partial derivatives happen to exist.
Precisely, f is differentiable at a point a if there is a linear map L (which must then be the total derivative) such that f(a + h) - f(a) - L(h) is o(|h|) as h -> 0 — the leftover error, divided by the length |h| of the step, goes to zero no matter the direction of h. Two facts make this the right definition. First, differentiability implies continuity and implies all directional derivatives behave linearly (the directional derivative in direction u is just the gradient dotted with u). Second, it is strictly stronger than having partials: a function can possess both partial derivatives at a point and still not be differentiable there, because the approximation is good only along the axes and fails on slanted approaches.
The workhorse sufficient condition is the one you will actually use: if all the partial derivatives exist and are continuous in a neighborhood of the point (such a function is called continuously differentiable, or C^1), then f is differentiable there. Since almost every formula built from polynomials, exponentials, sines, and logs is C^1 wherever it is defined, differentiability is usually automatic — you only stop to think at patched points, corners, or places where a denominator vanishes.
Let f(x, y) = xy / (x^2 + y^2) with f(0, 0) = 0. Both partials at the origin are 0, so a naive student declares it differentiable. But f is not even continuous at (0, 0), so it cannot be differentiable — proof that existing partials are not enough.
Partials can both exist and vanish while the function is discontinuous — differentiability rules this out.
Continuous partials are sufficient but not necessary for differentiability. A function can be differentiable at a point where its partials are not continuous (a standard but rare example), so C^1 is a convenient shortcut, not the definition.