Partial Derivatives: Definition & Computation
One variable freed, the rest frozen — the ordinary derivative reborn in higher dimensions
1. The Definition — A Promise Kept
The Core Idea
In single-variable calculus you learned the derivative as a limit of a difference quotient:
$$ f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} $$Now suppose the function depends on two (or more) variables: $z = f(x,y)$. We want to ask: how fast does $f$ change when we nudge $x$ alone, keeping $y$ frozen?
Callback: Compare these with the single-variable difference quotient — the only change is that one variable stays put while the other moves by $h$. The limit machinery is identical.
Notation — Two Traditions
| Style | Partial w.r.t. $x$ | Partial w.r.t. $y$ | How to read it |
|---|---|---|---|
| Leibniz (rounded-d) | $\dfrac{\partial f}{\partial x}$ | $\dfrac{\partial f}{\partial y}$ | "the partial of $f$ with respect to $x$" |
| Subscript | $f_x$ | $f_y$ | "$f$ sub $x$" |
| At a point | $f_x(a,b)$ or $\left.\dfrac{\partial f}{\partial x}\right|_{(a,b)}$ | $f_y(a,b)$ | "the partial of $f$ with respect to $x$ at $(a,b)$" |
The symbol $\partial$ (a rounded "d") is not the same as $d$; it signals that other variables are being held constant.
Value at a Point vs. the Partial-Derivative Function
Just as in single-variable calculus we distinguish between $f'(a)$ (a number) and $f'(x)$ (a function), we distinguish:
- $f_x(a,b)$ — a single number, the slope at the specific point $(a,b)$.
- $f_x(x,y)$ — a new function of two variables, giving the slope at every point where the limit exists.