---
title: "Partial Derivatives"
description: "Differentiate with respect to one variable while holding the others fixed. Mixed second partials agree for the functions met in practice, which is a statement worth noticing rather than assuming."
canonical: https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9
source: https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9.md
retrieved: 2026-09-12
---

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# Partial Derivatives

Differentiate with respect to one variable while holding the others fixed. Mixed second partials agree for the functions met in practice, which is a statement worth noticing rather than assuming.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9

## Ready when they can

- Compute both first partials of a function of two variables
- Interpret a partial derivative as a slope along a coordinate direction
- Check that the two mixed second partials agree for a given function

## Lesson: Slopes in the x and y directions

A partial derivative measures slope in one coordinate direction while the other variable stays frozen. To get fx, treat every y as a constant and differentiate in x. For x squared + 3xy + y squared, fx = 2x + 3y, so fx(2, 1) = 7.

**Example.** To get fy, freeze x instead. For the same f, fy = 3x + 2y, so fy(2, 1) = 8. Try g = 5x squared + 4xy minus 7y cubed: gy = 4x minus 21y squared, so gy(1, 1) = minus 17.

Geometrically, fx(a, b) is the slope of the trace you cut by holding y fixed at b. That trace is an ordinary single variable curve. Ana is right to picture fx as its slope.

**Tip.** Second partials differentiate twice, and for smooth functions the mixed orders agree. For x squared y cubed, fx = 2x y cubed and fxy = 6x y squared gives 24 at (1, 2). If your fxy and fyx ever differ, recompute the first partials.

**Recap.** Freeze one variable, differentiate in the other, and let matching mixed partials confirm your algebra.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [The Product and Quotient Rules](https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv)
- [Functions of Several Variables and Level Curves](https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8)

## Opens up

- [Tangent Planes and Linear Approximation](https://lightmysky.com/learn/mathematics/tangent-planes-and-linear-approximation-mt_EYMcjChX_m)
- [The Euler-Lagrange Equation and Conserved Momenta](https://lightmysky.com/learn/science/the-euler-lagrange-equation-and-conserved-momenta-mt_GSEFABbEXI)
- [Classifying Second-Order PDEs and What the Type Decides](https://lightmysky.com/learn/mathematics/classifying-second-order-pdes-and-what-the-type-decides-mt_hlQWbd1kzR)
- [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss)
- [Chemical Potential and Gibbs Energy That Depends on Composition](https://lightmysky.com/learn/science/chemical-potential-and-gibbs-energy-that-depends-on-composition-mt_Tdl6K-gFO3)