--- title: "Partial Derivatives" description: "Differentiate with respect to one variable while holding the others fixed. Interpret coordinate slopes and compare mixed second partials, which agree when both mixed derivatives are continuous near th" canonical: https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9 source: https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Partial Derivatives Differentiate with respect to one variable while holding the others fixed. Interpret coordinate slopes and compare mixed second partials, which agree when both mixed derivatives are continuous near the point. 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 - Compute mixed second partials in both orders and check their agreement under continuity conditions ## Lesson: Slopes in the x and y directions For f(x, y), fx means differentiate in x with y fixed. For a positive integer n, the x derivative of x^n is n x^(n-1). Keep fixed factors as coefficients. Constants have derivative 0, so y-only terms contribute 0 to fx. For h = 2x² + 5xy + y³, hx = 4x + 5y, so hx(1, 2) = 14. You may substitute the fixed y-value early, but keep x variable until you have differentiated. To find fy, hold x fixed and differentiate in y. Terms containing only x have derivative 0 in y. For h = 2x² + 5xy + y³, hy = 5x + 3y², so hy(1, 2) = 17. A fixed x-factor acts as a coefficient: the y derivative of x²y³ is 3x²y². Differentiate before substituting the requested y-value. On z = f(x, y), fixing y = b leaves the trace z = f(x, b). Its tangent slope at x = a is fx(a, b): change in z per change in x. Fixing x = a instead leaves g(y) = f(a, y), whose tangent slope at y = b is g'(b) = fy(a, b). Positive and negative slopes mean that the tangent line rises or falls in the direction of the increasing input. A second partial differentiates a first partial. Here fxy means differentiate in x, then y; fyx means y, then x. Keep the first partial as a function until both differentiations are finished. For h = 3x²y², hx = 6xy² and hxy = 12xy. In the other order, hy = 6x²y and hyx = 12xy. Evaluate at the point after finding the mixed derivative. If fxy and fyx exist and are continuous on an open neighborhood of a point, they agree there. A smooth function has continuous derivatives of every order on an open domain, so its mixed partials agree at every point of that domain. A mismatch means at least one first or second differentiation needs checking. Agreement alone does not prove the first partials are correct: errors can disappear during the second differentiation. A zero partial derivative at one point does not make the whole trace constant. For h = x² + y², hx(0, 2) = 0. Yet the trace h(x, 2) = x² + 4 gives 4 at x = 0 and 5 at x = 1. Its tangent is horizontal at x = 0, but its values are not constant. The fixed-variable rule also applies to trigonometric terms. Use radians: the y derivative of sin y is cos y. With x fixed, the y derivative of x sin y is x cos y. For h = x sin y + 2y², hy = x cos y + 4y. At (3, 0), hy = 3 because cos 0 = 1. *(drawing: The same formula gives different derivatives because a different input stays fixed.)* **Recap.** Hold the other input fixed, differentiate, then evaluate. Continuous mixed partials agree, but agreement alone does not prove that every step is correct. *Lesson, questions and flashcards written for this learning dataset. OpenStax Calculus Volume 3 and Khan Academy materials were studied for mathematical facts and teaching structure.* ## Practice 16 questions on this page, each with its working shown. ## Needs first - [Functions of Several Variables and Level Curves](https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8) ## Optional connections - [The Product and Quotient Rules](https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv) ## 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)