Partial derivatives: slicing the surface

"What's the slope of this surface?" is a trick question — standing on a hillside, the slope depends on which way you face. Multivariable calculus answers with a strategy, not a number: face one axis at a time.

Freeze and slice

The partial derivative of f with respect to x, written ∂f/∂x, is the ordinary derivative you get by treating y as a frozen constant. Geometry: slice the surface with a vertical plane running in the x-direction; the cut edge is a plain single-variable curve, and ∂f/∂x is its slope.

Two variables, two slice directions, two partials. That curly ∂ (say "partial") is just a reminder that the other variable is being held still.

Computing: everything you know still works

Differentiate with respect to x pretending y is a number like 7:

$f(x, y) = 4 - \tfrac{x^2}{2} - y^2 \quad\Rightarrow\quad \frac{\partial f}{\partial x} = -x, \qquad \frac{\partial f}{\partial y} = -2y$

At the point (1, 1): ∂f/∂x = −1 and ∂f/∂y = −2 — walking east you descend at slope 1, walking north you descend twice as fast. All your derivative rules — power, product, chain — carry over untouched.

Read them like a dashboard

Partials are sensitivity readings: if I nudge this one input, how fast does the output respond? A pricing model's ∂revenue/∂price, a physics field's ∂T/∂x, a loss function's ∂L/∂weight (the atom of machine learning) — each is a partial derivative with a job.

Check yourself in the surface explorer: it prints ∇f — the pair of partials — for any family at any point, computed exactly.

Why this matters

Two numbers can't be the whole story — you can face infinitely many directions, not just two. The next lesson glues the partials into a single arrow that answers every direction at once.