Foundations — Partial derivatives — notation, calculation, geometric meaning

This page assumes nothing. If the parent note wrote a symbol, we build it here first, in order, each brick resting on the one before.

1. A function of one variable — the thing you already know

The picture is a curve on flat paper. The horizontal axis is the input $x$; the vertical axis is the output $y=f(x)$. Every input pins down a single height.

Why the topic needs this: partial derivatives are secretly ordinary one-variable derivatives. If you can't picture $f(x)$ as a curve, the whole "freeze and differentiate" trick has nowhere to stand.

2. Slope — the number a derivative produces

$\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{\text{change in output}}{\text{change in input}}$

The picture is a straight line just kissing the curve at one point — the tangent line. Its steepness is the slope there.

Why the topic needs this: a partial derivative is defined as the slope of a tangent line. Slope is the currency the whole subject trades in.

3. The limit ${\lim }_{h\to 0}$ — "let the step shrink to nothing"

Why not just plug $h=0$? Because slope is $\frac{\text{rise}}{\text{run}}$, and at $h=0$ the run is $0$ — division by zero, meaningless. So we shrink the run toward $0$ and watch what the ratio approaches.

Why the topic needs this: the parent's very definition is $\frac{\partial f}{\partial x}(a,b)={\lim }_{h\to 0}\frac{f(a+h,b)-f(a,b)}{h}.$ You cannot read that line without knowing ${\lim }_{h\to 0}$ means "shrink $h$ to nothing".

4. The single-variable derivative $f^{\prime }(x)$

The three rules you must have ready (proved in Single-variable derivative):

Why the topic needs this: the entire computation method is "freeze the other variable, then use exactly these rules." No new calculus is invented — only reused.

5. Two inputs at once — the surface $z=f(x,y)$

The picture is no longer a curve on paper — it needs 3D space. Lay $x$ and $y$ flat like a map (the floor), and let $z$ be height above the floor. Every point $(x,y)$ on the floor gets a height, so the outputs form a surface — a landscape of hills and valleys.

Why the topic needs this: the parent's whole problem — "a surface has infinitely many directions, so no single slope" — only makes sense once you see $f(x,y)$ as a surface, not a curve.

6. The point $(a,b)$ and the pair notation

So $f(a,b)$ is the single height directly above that spot.

Why the topic needs this: every evaluated partial is written at a point, e.g. $f_x(1,2)=2$. The pair $(a,b)$ is the address; without it "the slope here" has no here.

7. Freezing a variable — turning a surface back into a curve

This is the pivotal move, so we build it slowly.

Freezing $y=b$ and letting only $x$ move defines a brand-new one-variable function: $g(x)=f(x,b).$ Its graph is exactly that slice-curve — an ordinary curve on paper again. And now everything from Sections 2–4 applies to $g$.

Why the topic needs this: this is why partial derivatives are ordinary derivatives in disguise. The parent's key line $\frac{\partial f}{\partial x}(a,b)=g^{\prime }(a)$ is nothing but "the partial is the slope of the sliced curve at $x=a$."

8. The curly $\partial$ — a new symbol for a new caution

The parent lists these as identical shorthands: $\frac{\partial f}{\partial x}=f_x={\partial }_{x}f=D_{x}f.$ All four mean the same number. Read $f_x$ as "eff-sub-ex" — the subscript names the variable that moved.

Why the topic needs this: every formula in the parent uses $\partial$ or $f_x$. This section is where those glyphs earn their meaning.

9. Subscripts stacked — higher and mixed partials

The picture: $f_{xx}$ is the curvature of a slice (how the slope bends); $f_{xy}$ is the surface's twist — how the East slope tilts as you walk North. That twist symmetry is Clairaut's theorem.

Why the topic needs this: the parent's Clairaut section ($f_{xy}=f_{yx}$) is unreadable without knowing how stacked subscripts encode "differentiate again, in this order."

10. Where these feed — the tangent plane preview

You don't need to derive this here — that's Tangent plane and linear approximation. Just recognise the ingredients: $f(a,b)$ is the height (Section 6), $f_x(a,b)$ and $f_y(a,b)$ are the two slice-slopes (Sections 7–8), and $(x-a),(y-b)$ are how far you've walked from the spot. Two slopes in two directions pin down a flat sheet — the tangent plane.

Prerequisite map

Equipment checklist

Test yourself — you should be able to answer each before moving on.

Connections

• Parent topic — the note these foundations feed.
• Single-variable derivative — Sections 1–4 live here in full.
• Tangent plane and linear approximation — where the two slopes combine.
• Gradient vector — packages $f_x,f_y$ together.
• Directional derivative — generalises to any walking direction.
• Clairaut's theorem — the twist symmetry of Section 9.
• Chain rule (multivariable) · Total differential — next steps.