Partial Derivatives

A partial derivative asks how f changes when one input moves and the other is held fixed. Freezing a variable turns a hard problem into an ordinary one.

Notation

The partial of f with respect to x is written fₓ or ∂f/∂x. The curly ∂ signals that other variables are being held still.

How to compute

Treat y as a number and differentiate normally. Everything you learned about products, quotients and chains still applies.

What it means on the surface

Holding y fixed slices the surface into a curve. The partial derivative is the ordinary slope of that curve.

Second partials

Differentiating twice with respect to x gives fₓₓ. Differentiating once each way gives the mixed partial fₓᵧ.

Clairaut's theorem

For the functions you will meet, fₓᵧ equals fᵧₓ. The order of mixed differentiation does not matter.

Hold one variable still

The partial derivative with respect to x treats y as a constant and differentiates as usual. Every single-variable rule applies unchanged, which makes computation easy once the idea is accepted.

It is a slope in one direction

Slicing the surface with a plane of constant y gives a curve, and the partial derivative is its slope. Each partial measures the rate of change along one axis only.

The curly d

∂f/∂x rather than df/dx signals that other variables are being held fixed. The distinct symbol exists because the two are genuinely different operations.

Mixed partials usually commute

Differentiating with respect to x then y gives the same result as y then x, provided the second partials are continuous. That is Clairaut's theorem, and it is a useful check on computation.