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

Graphical or geometric meaning

Intersect the surface $z=f(x,y)$ with the vertical plane $y=y_0$; the tangent slope of that trace is $f_x$. The plane $x=x_0$ produces the trace whose slope is $f_y$.

Common mistakes and why they fail

Verification and reasonableness checks

• Treat the expression as a one-variable function and use a difference quotient.
• Compare symbolic partials with trace slopes.
• Track subscripts in the order actually differentiated.

“Hold fixed” is a local instruction

The partial derivative $f_x(a,b)$ measures change in $x$ while $y$ is held at $b$. Compute it by treating the other variables as constants, then evaluate at the point. Mixed partials often agree when the relevant second partials are continuous near the point, but equality is not automatic without hypotheses. Existence of all partial derivatives at one point also does not guarantee continuity or differentiability there. Units clarify interpretation: $f_x$ has output units per $x$ unit, whereas $f_y$ uses the $y$ unit. After differentiating, compare with a one-variable slice such as $g(x)=f(x,b)$. This slice check catches terms that were mistakenly differentiated through a supposedly fixed variable.

Practice

1. Find $f_x$ for $f=x^3y+y^2$.
2. Find $f_y$ for the same function.
3. What is held fixed when computing $f_x(x,y)$?

Connections and next steps

• Gradient
• Tangent Planes
• Chain Rule in Several Variables