From the line to R^n
In one-variable calculus the input lives on the real line and the derivative measures one slope. Now the input is a point x = (x_1, ..., x_n) in R^n, and a function of several variables f: U -> R takes such a point to a single number. The domain U is an open subset of R^n, so every point of U has a small ball around it — a neighborhood — sitting entirely inside U. That room to move in every direction is exactly what we will need to talk about derivatives.
A handy picture is the level set {x : f(x) = c}: for f(x, y) = x^2 + y^2 these are circles, and the graph is a bowl. Limits and continuity are defined just as on the line, but now |x - a| is the Euclidean distance, so x -> a means the point approaches a from all directions at once.
Partial and directional derivatives
The simplest derivative freezes all variables but one. The partial derivative of f with respect to x_j at a is the ordinary one-variable derivative of t -> f(a + t e_j), where e_j is the j-th coordinate direction. More generally, for a unit vector v the directional derivative is the limit below, the rate of change as you walk away from a in direction v.
Partial derivative of f at a in direction e_j:
D_j f(a) = lim_{t -> 0} ( f(a + t e_j) - f(a) ) / t
Directional derivative in unit direction v:
D_v f(a) = lim_{t -> 0} ( f(a + t v) - f(a) ) / t
Worked example: f(x, y) = x^2 y, a = (1, 3).
Hold y fixed: d/dx ( x^2 * 3 ) = 6x, at x=1 gives D_1 f(a) = 6.
Hold x fixed: d/dy ( 1 * y ) = 1, so D_2 f(a) = 1.
Now a diagonal direction v = (1/sqrt2, 1/sqrt2):
g(t) = f(1 + t/sqrt2, 3 + t/sqrt2)
g'(0) = 6 * (1/sqrt2) + 1 * (1/sqrt2) = 7/sqrt2.
Notice 7/sqrt2 = (6, 1) . (1/sqrt2, 1/sqrt2) — partials predicted it.Why partials are not enough
The lesson: partials only probe the coordinate axes, and even all directional derivatives can exist without f being continuous. We need a stronger condition that controls f along every path simultaneously. That is the total derivative, the subject of the next guide.