Partial Derivatives

The central idea

For $f:\mathbb R^n\to\mathbb R$, the partial derivative $f_{x_i}(\mathbf a)$ is the one-variable derivative obtained by varying $x_i$ while holding other coordinates fixed. Second partials include pure terms $f_{x_ix_i}$ and mixed terms $f_{x_ix_j}$. If mixed second partials are continuous near a point, Clairaut's theorem gives equality of their order.

Definitions, hypotheses, and notation

Partial derivative notation can be operator-based, $\partial f/\partial x$, or subscript-based, $f_x$. In $f_{xy}$, conventions may differ about reading order, so writing the intermediate derivative removes ambiguity. For $C^2$ functions the mixed values agree and the convention becomes harmless.

A function can have every directional or coordinate partial derivative at a point and still fail to be continuous there. Differentiability requires one linear map to approximate all simultaneous small changes, which is stronger than separate coordinate-line behavior.

Conceptual meaning

A partial derivative is slope along a coordinate line through the input point. It measures sensitivity to one input under a ceteris paribus assumption. The collection of first partials forms the gradient, but partials alone need not guarantee differentiability.

A dependable method and decision rule

1. Identify the differentiation variable and explicitly freeze the others.
2. Apply one-variable differentiation rules to the resulting expression.
3. Retain the other coordinates as constants, not as zero.
4. For higher partials, differentiate the already obtained partial in the stated order.
5. Check domain and continuity before invoking mixed-partial symmetry.

Fully worked example

Common mistakes and why they fail