Module 7 · Lesson 7.1

Partial Derivatives — Slicing Multivariable Functions

Freeze every variable but one to measure a multivariable function's slope along a single direction at a time.

Intuition

Standing on a hillside, "how steep is it here?" doesn't have one answer — walking north might be a gentle climb while walking east is a steep drop. A function of two variables, like terrain height $h(x, y)$, works the same way: there's no single slope, only a slope in each specific direction you might walk.

Core concept

A partial derivative answers "how steep, in exactly this one direction?" by freezing every other variable as if it were a fixed constant, then differentiating normally. $\frac{\partial h}{\partial x}$ treats $y$ as locked in place and measures the slope as $x$ alone changes; $\frac{\partial h}{\partial y}$ does the reverse, freezing $x$ and measuring the slope as $y$ alone changes.

slice along x, y frozen
Terrain height functionh(x,y) = 10 − 0.5x² − 0.3y²
∂h/∂x (freeze y)−x
∂h/∂y (freeze x)−0.6y
At point (2, 3): ∂h/∂x−2
At point (2, 3): ∂h/∂y−0.6(3) = −1.8
Answersteeper descent along x (−2) than along y (−1.8)

Interactive Lab

The terrain's $y$-coordinate is frozen at 2. Slide $x$ to move along that single slice of the hill, and watch the tangent line show the slope, $\partial h/\partial x$, at that exact spot.

x h
x = 0.0 h(x, 2) = 8.80 ∂h/∂x = 0.0

Key Idea

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

This is the ordinary derivative limit, with one change: $y$ never moves. Only $x$ gets nudged by the tiny step $h$, so whatever slope comes out reflects movement in the $x$-direction alone.

$\partial f/\partial x$The partial derivative — slope of $f$ in the $x$-direction only
$f(x+h,y)$The function's value after a tiny nudge in $x$ alone
$h$A tiny step size in the $x$-direction (not to be confused with terrain height)
$y$ held fixedEvery other variable stays frozen while the limit is taken

Think Further

  1. Could $\partial h/\partial x$ be positive while $\partial h/\partial y$ is negative at the exact same point on a terrain map? What would that say about the shape of the terrain there?
  2. Predict: standing at the very peak of a hill, like the origin in this lesson's terrain function, what would you expect both $\partial h/\partial x$ and $\partial h/\partial y$ to equal there, and why?
  3. A partial derivative only tells you the slope along one specific axis. What information would you be missing if you wanted to know the single steepest direction to walk, rather than just the slope along x or along y?
Show suggested answers
  1. Yes — the terrain could be rising as you move in the x-direction while simultaneously falling as you move in the y-direction at that same point, which happens naturally on a tilted or saddle-shaped surface.
  2. Both should equal zero. At the very peak, the terrain is momentarily flat in every direction since it's a local maximum, so the tangent slice in either single direction has no slope there.
  3. You'd be missing the slopes in every direction besides the x- and y-axes. Partial derivatives alone don't reveal which combined direction gives the steepest climb — that requires combining them into a single gradient vector, which is exactly the next step.