Foundations — Chain rule for multivariable functions — all cases

Before you can read the parent note, you need a small toolbox. This page builds every symbol it uses, starting from nothing. Read top to bottom — each item leans on the one before it.

0. What is a function of one input? ($y=f(x)$)

Figure s01 — what to look at: a literal box with a knob labelled $x$ on the left and a dial labelled $y$ on the right. Turn the input knob and the output dial swings. The box carries the name $f$ — the machine's name.

The letter $f$ is just a name for the machine. If we had a second machine we might call it $g$. The parent note uses $f$ for the main output machine and $g$ for an input machine.

1. Rate of change: the ordinary derivative $\frac{dy}{dx}$

Now: how fast does the output move when you nudge the input? That is the whole point of calculus.

Figure s02 — what to look at: the blue curve is $y=f(x)$; the orange straight line just kisses it at the red point. The little green right-triangle shows a tiny run $dx$ along the bottom and the tiny rise $dy$ up the side. The slope of the orange line is $dy$ divided by $dx$.

Why the topic needs it. The chain rule multiplies rates of change "link by link". Each link is exactly one of these ratios. If you cannot read $\frac{dy}{dx}$ as "how much output per unit input", nothing downstream makes sense.

See Single-variable Chain Rule for the one-input version of the whole idea.

2. Functions of two inputs: $z=f(x,y)$

Real machines often have two knobs. Write $z=f(x,y)$: two inputs $x$ and $y$, one output $z$.

Figure s03 — what to look at: instead of a flat curve we now have a whole surface hovering over the floor. Any point on the floor is named by two numbers $(x,y)$ (east and north); the height of the surface straight above it is $z$. The red dot and its grey drop-line show one such point $(x,y,z)$.

Why the topic needs it. The parent note says an output "has two doors" $x$ and $y$. Those doors are exactly the two input slots. A wiggle can come in either door.

3. Partial derivative $\frac{\partial z}{\partial x}$ — the slanted-d

With two knobs, "the" slope is ambiguous — slope in which direction? So we ask a narrower question: freeze one knob, wiggle the other.

Figure s04 — what to look at: we pick one fixed value of $y$ and slice the surface with a wall running in the $x$-direction. The orange curve is the exposed edge of that slice — a one-input curve again. The red tangent line's slope on this slice is exactly $\frac{\partial z}{\partial x}$. On this surface at the marked point the slice climbs to the left of the crest, so the slope shown is positive (uphill in the $+x$ direction).

Why the topic needs it. Every "link" in a multivariable chain that starts at $z=f(x,y)$ is a partial derivative: $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ are the two rates one per door.

Full detail lives in Partial Derivatives.

3b. When to use $d$ and when to use $\partial$ — reconciled

This trips everyone, so read slowly. The choice depends on how many variables the thing on top ultimately depends on, counting all the way down the chain.

4. The total differential $dz$ — combining both doors

Now nudge both knobs a tiny bit at once. How much does $z$ change in total?

Figure s05 — what to look at: the blue curved surface with its flat orange tangent plane resting on it at the red point. The green step $dx$ east lifts you by $z_{x}dx$; the green step $dy$ north lifts you by $z_{y}dy$; together the plane lifts you by their sum $dz$. The tiny gap between the plane and the true surface as you step away is the error $\epsilon$ that vanishes faster than the step.

From the total differential to the chain rule — the limit written out. Suppose the two knobs are driven by a single deeper variable $t$ (a deep cause such as time; its tiny nudge is $dt$ and its measurable nudge is $\Delta t$). Start from the exact change from the differentiability box and divide every term by the measurable step $\Delta t$:

$\frac{\Delta z}{\Delta t}=\frac{\partial z}{\partial x}\frac{\Delta x}{\Delta t}+\frac{\partial z}{\partial y}\frac{\Delta y}{\Delta t}+{\epsilon }_1\frac{\Delta x}{\Delta t}+{\epsilon }_2\frac{\Delta y}{\Delta t}.$

Why this is legal: dividing a sum by $\Delta t$ divides each piece separately. Now let $\Delta t\to 0$. Because $x$ and $y$ move continuously with $t$, the steps $\Delta x,\Delta y\to 0$ too, which forces ${\epsilon }_1,{\epsilon }_2\to 0$. Meanwhile the ratios stay finite: $\frac{\Delta x}{\Delta t}\to \frac{dx}{dt}$ and $\frac{\Delta y}{\Delta t}\to \frac{dy}{dt}$. So the last two terms are (something $\to 0$) $\times$ (something finite) $\to 0$ and drop out entirely:

$\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}.$

That is the whole chain rule for one deep variable, and now you have watched the error term die instead of taking it on faith. Note $z$ finally depends on the single variable $t$, so the straight $d$ is correct in $\frac{dz}{dt}$ (§3b). See Total Differential.

5. Composition: inputs that depend on deeper things

Here is the twist that makes it multivariable chain rule and not just partials.

The dependency tree is a picture of this "who-feeds-whom": the output sits on top, and arrows point downward to whatever each thing depends on. The map below is that same tree for this whole page — read it as a story of how each foundation feeds the next.

How to read the map: start at the top box y = f(x) — the simplest machine. The arrow to dy/dx says "to understand one input you first need its rate of change". That feeds into the two-input surface z = f(x,y), which needs the partial derivative (freeze one knob), which lets us write the total differential (nudge both, add). Divide that by a deep variable and you reach composition, and finally the Multivariable Chain Rule box at the bottom. Two side-boxes — the summation sign and the Jacobian — feed straight into that final box because the general chain rule needs both. Follow any arrow and read it as "you need the box behind the arrow before the box in front of it".

Why the topic needs it. Without nesting there is no chain to follow. The parent's cookie story (rain → flour & sugar → cookies) is exactly this: $t$ is rain, $x,y$ are flour and sugar, $z$ is cookies.

6. The summation sign $\sum$ and the index $i$

When there are many inputs $x_1,x_2,\dots ,x_n$ the parent note packs the sum into one symbol.

Why the topic needs it. Writing every term out is exhausting when there are $n$ doors; ${\sum }_{i=1}^n$ says the same in tiny space, and this is the exact form the parent's general rule uses.

7. Gradient and Jacobian — the compact packaging

Collect the partial derivatives into a single object so the whole chain rule becomes one multiplication.

You only need to recognise these for the parent's "matrix form" remark. Deep dive: Gradient and Jacobian. They also power the Directional Derivative and Implicit Differentiation (multivariable) connections the parent lists.

Equipment checklist

Test yourself — cover the right side. If any answer is a blank, reread that section before the parent note.

Connections

• Single-variable Chain Rule — the one-path warm-up.
• Partial Derivatives — every link in the chain is one of these.
• Total Differential — the seed line we divide by $dt$.
• Directional Derivative — the polar chain rule is one of these.
• Gradient and Jacobian — the compact matrix packaging.
• Implicit Differentiation (multivariable) — uses the explicit-term chain rule.