Multivariable functions & partial derivatives
The idea
Real models rarely depend on one input: temperature varies with position and time, revenue with price and volume. A function f(x, y) defines a surface over the plane, and partial derivatives are the natural extension of f'(x): the partial ∂f/∂x differentiates with respect to x while y is held frozen, and ∂f/∂y does the reverse. Each one is an ordinary single-variable derivative along one axis direction, so every rule you know — product, chain, power — carries over unchanged.
Picture standing on the surface at a point: ∂f/∂x is the slope you feel walking due east, ∂f/∂y the slope walking due north. The pair (f_x, f_y) summarizes the local terrain to first order. The misconception to avoid: treating the other variable as a constant is not an approximation or a simplification trick — it is the definition. Each partial deliberately measures sensitivity to one input alone, and combining sensitivities across directions is a separate job (the gradient and the total differential handle it).
Worked example
Let f(x, y) = x²y³ − 4xy. Compute both partial derivatives, then evaluate them at the point (2, 1) and interpret the results.
- For ∂f/∂x, freeze y and differentiate in x: the term x²y³ has constant coefficient y³, giving 2xy³, and −4xy gives −4y. So f_x = 2xy³ − 4y.
- For ∂f/∂y, freeze x: x²y³ differentiates to 3x²y², and −4xy to −4x. So f_y = 3x²y² − 4x.
- Evaluate at (2, 1): f_x(2, 1) = 2(2)(1) − 4(1) = 4 − 4 = 0, and f_y(2, 1) = 3(4)(1) − 4(2) = 12 − 8 = 4.
- Interpret: at (2, 1) the surface is momentarily flat in the x-direction — an east-west walk neither climbs nor falls to first order — while a step north raises f at 4 output units per unit of y.
- Cross-check f_x numerically: f(2.01, 1) = 4.0401 − 8.04 = −3.9999 versus f(2, 1) = −4, a change of 0.0001 over a step of 0.01 — a measured slope of 0.01, consistent with an exact first-order slope of 0.
Answer. f_x = 2xy³ − 4y and f_y = 3x²y² − 4x; at (2, 1) they equal 0 and 4 respectively.
Check your understanding
- Why is holding the other variable constant part of the definition of a partial derivative rather than an approximation?
- What does the pair of slopes (f_x, f_y) at a point let you predict about small moves in arbitrary directions?
- How would you check a computed partial derivative numerically, and which step-size pitfalls should you watch for?
- What does it mean about the surface when both partials vanish at a point, and why is that not enough to declare a local minimum?
Build the foundations first
Multivariable functions & partial derivatives builds on these concepts. If any feel shaky, start there.