Partial derivatives

For a function of one variable, "the slope" is unambiguous — there is only one direction to move. On a surface z = f(x, y) that stops being true. Standing on a hillside you can walk east and climb, walk north and descend, or walk along the contour and stay level. A single number cannot capture that, so the derivative splits into several.

A partial derivative fixes all variables but one and differentiates as usual. ∂f/∂x asks how f changes as x moves with y frozen; ∂f/∂y asks the reverse. Computationally this is easy — treat the other variable as a constant — but the word "constant" is doing work that gets misread. Constants do not vanish, they stay as coefficients. For f = x²y, differentiating with respect to x gives 2xy, not 2x: y is held fixed, not deleted.

The two partials together give the gradient, which points in the direction of steepest ascent and whose length is that steepest slope. From it you can get the rate of change in any direction, so two numbers do describe the whole local picture. Finding where both partials vanish locates the flat points, but this generalises the one-variable story in an important way: as well as maxima and minima, a surface can have a saddle point, rising in one direction and falling in the perpendicular one. A mountain pass is exactly that, and no amount of staring at first derivatives distinguishes it — the second derivatives have to be examined.