Partial derivatives: hold every other variable constant
Assistance allowed only after prediction and observed evidence are recorded.
Prerequisites
Day 19 chain rule drills; if the scalar rule is shaky, open the day-level recovery link before continuing.
Learning Objective
Define the partial derivative as the ordinary derivative along one axis with all other variables frozen, compute both partials of a polynomial by hand, and confirm them with finite differences.
Instructions
Study source: within this activity's existing reading/practice time, use MIT OCW 18.02SC Multivariable Calculus, Part 2: Partial Derivatives; differentiation, chain rule and gradient sessions, then explain one connection to the local task in your own words. Keep the local exercise and acceptance checks; use its original explanation if the external source is unavailable. For f(x, y) = x^2 y + 3y - x, freeze y and differentiate in x: ∂f/∂x = 2xy - 1. Freeze x and differentiate in y: ∂f/∂y = x^2 + 3. At the point (2, 1): f = 4*1 + 3 - 2 = 5, ∂f/∂x = 4 - 1 = 3, ∂f/∂y = 4 + 3 = 7. Predict these before checking with central differences: (f(2.00001, 1) - f(1.99999, 1)) / 0.00002 approaches 3, and the y-slice approaches 7.
Practice & Prediction Log
Compute by hand and verify numerically: (a) both partials of f(x, y) = x^2 y + 3y - x at (2, 1); (b) both partials of f(x, y) = x y^2 at (3, 2). Record the frozen variable for every differentiation.
For (b), ∂f/∂x treats y^2 as a constant coefficient on x, giving y^2; ∂f/∂y treats x as a constant coefficient, giving 2xy.
(a) ∂f/∂x = 2xy - 1 = 3 and ∂f/∂y = x^2 + 3 = 7 at (2, 1); f(2, 1) = 5. Central differences return about 3.0000 and 7.0000. (b) f(x, y) = x y^2: ∂f/∂x = y^2 = 4 and ∂f/∂y = 2xy = 12 at (3, 2); f(3, 2) = 12. If your symbolic result and the finite difference disagree, check all three: the derivation (which variable you held fixed on each line), the numerical check code (function definition, evaluation point, which variable is perturbed), and the step size h (compare h = 1e-3, 1e-4, 1e-5; a true match stays stable).
You compute both partials for both functions unaided, state which variable was frozen in each line, and your four values 3, 7, 4, 12 all match a finite-difference check you ran yourself.
Authoritative Source Resources & Provenance
MIT OCW 18.02SC Multivariable Calculus
Relevant Section / Range: Part 2: Partial Derivatives; differentiation, chain rule and gradient sessions
Verified Version & Date: Fall 2010 (Verified: 2026-10-30)
Rigorous university-lecture treatment that lets motivated learners go deeper on partial derivatives and the gradient after the OpenStax primary text.
Single-variable calculus and coordinate geometry
All required formulas, worked examples, and numeric checks are contained in the localized lesson text.
Compared Candidate Sources & Selection Analysis (2)
Chosen as the primary written reference; MIT OCW is the deeper lecture companion.
More accessible pacing but less rigorous than the Auroux lectures for the path-sum chain rule.
OpenStax Calculus Volume 3
Relevant Section / Range: Section 4.3 Partial Derivatives: definition, notation, and worked examples
Verified Version & Date: OpenStax Calculus Volume 3 web edition (verified 2026-10-30)
Openly licensed primary definition of the partial derivative matching the day's slice-and-differentiate method.
Single-variable derivatives and the chain rule (Study Days 16-19).
The lesson contains complete local derivations, worked answers, and a finite-difference check.
Compared Candidate Sources & Selection Analysis (2)
Selected as the lecture-depth companion for partial derivatives; OpenStax Volume 3 remains the written primary reference.
Strong visual coverage of gradients, but the OpenStax text better matches the lesson's theorem-plus-hand-check structure.