Calc III · §14.1 partial derivatives

A partial derivative is the slope of a slice.

Slice the surface z = f(x,y) with the vertical plane y = b. The cut is a curve C₁ through the point P=(a,b,c). Flattened, C₁ is just a one‑variable function g(x)=f(x,b) — and fx(a,b)=g'(a) is the slope of its tangent line, the one pointing along the x‑axis. Fix x = a instead and you get C₂, whose tangent slope is fy(a,b). Drag the sliders to move the cutting planes; drag the graph to rotate.

Two slices, two slopes: ∂f/∂x holds y fixed, ∂f/∂y holds x fixed. 🏔️