Partial Derivatives

Summary

A partial derivative measures how a multivariable function changes when one variable varies and the others are held fixed. Geometrically, it is the slope of the curve obtained by slicing the graph with a plane of constant remaining variables.

Prerequisites

Derivatives, Multivariable Functions

Definition

For $f (x, y)$ ,

f_{x} (x, y) = \frac{\partial f}{\partial x} = lim_{h \to 0} \frac{f (x + h, y) - f (x, y)}{h},

when the limit exists, and similarly for $f_{y}$ .

Conditions / Assumptions

When computing $\partial f / \partial x$ , treat every other independent variable as constant.
Existence of partials at a point does not imply continuity or differentiability of $f$ there.

Worked Example

If $f (x, y) = x^{2} + 3 x y - 4 y^{2}$ , then

\frac{\partial f}{\partial x} = 2 x + 3 y, \frac{\partial f}{\partial y} = 3 x - 8 y .

If $f (x, y) = x^{2} y + 4 y^{3}$ , then $f_{x} = 2 x y$ and $f_{y} = x^{2} + 12 y^{2}$ .

The gradient packages the first partials:

\nabla f = (f_{x}, f_{y}) .

Common Mistakes

Differentiating with respect to $x$ while still treating $y$ as a function of $x$ unless the context is a total derivative (chain rule).
Confusing $\partial f / \partial x$ with the directional derivative in a non-axis direction.

Connections

Chain Rules, Directional Derivative, Differentiability of a Function, Clairaut's Theorem

References

Partial derivatives are introduced in OpenStax Calculus Volume 3.^[1]

OpenStax, Calculus Volume 3, Section 4.3, https://openstax.org/details/books/calculus-volume-3 ↩︎