Foundations — Chain rule — proof, composite function derivatives

Before you can read the parent note Chain Rule — Proof & Composite Function Derivatives, you need to own every symbol it throws at you. This page builds them one at a time, from absolutely nothing, in an order where each new idea leans only on the ones before it.

1. A function — a machine with one input and one output

The picture that makes this concrete: a box with an arrow going in (labelled $x$) and an arrow coming out (labelled $f(x)$).

The letters $f$, $g$, $h$ are just names for three different machines. There is nothing special about the letters — we could call them Machine-1, Machine-2, Machine-3. By habit:

• $g$ = the inner machine (runs first),
• $f$ = the outer machine (runs second),
• $h$ = the whole thing viewed as one combined machine.

2. Composition $f(g(x))$ — wiring two machines in a line

Read the figure left to right: $x$ enters $g$, out comes a middle number, that middle number enters $f$, out comes the final answer. The middle number is so important it gets its own name in the next section.

3. The letter $u$ — a name for "the middle number"

In the figure above, $u$ is the number travelling along the middle arrow. It is not a new mystery — it is literally $g(x)$ wearing a shorter name. The parent note writes "with $u=g(x)$ and $y=f(u)$"; now you know $y$ is just the final output and $u$ is the middle value.

4. $\Delta x$ — "a tiny nudge in the input"

Picture a point sitting at $x$ on a number line; $\Delta x$ is a short arrow pushing it a little to the right (or left, if $\Delta x$ is negative).

Two more deltas follow the same rule:

• $\Delta u=g(x+\Delta x)-g(x)$ — how much the middle number moved because of the nudge,
• $\Delta y=f(u+\Delta u)-f(u)$ — how much the final output moved.

5. The ratio $\frac{\Delta y}{\Delta x}$ — "steepness over a step"

Look at the figure: pick two points on the curve, one at $x$ and one at $x+\Delta x$. Draw the straight line joining them (the secant). Its slope is exactly $\frac{\Delta y}{\Delta x}$ — go along by $\Delta x$, go up by $\Delta y$.

6. The limit ${\lim }_{\Delta x\to 0}$ — "shrink the step to nothing"

Picture the secant line from the previous figure. Slide the second point toward the first: $\Delta x$ shrinks, the secant pivots, and it settles onto one special line — the tangent, the line grazing the curve at that single point. The limit is that settling-down.

7. The derivative $f^{\prime }(x)$ — "the exact steepness at a point"

The little tick mark ${}^{\prime }$ ("prime") just means "the derivative of." So $g^{\prime }(x)$ is the inner machine's stretch factor, and $f^{\prime }(u)$ is the outer machine's stretch factor at the middle value $u$ — note carefully, evaluated at $u$, not at $x$.

8. The two notations, side by side

Translation table:

• $\frac{dy}{du}$ = $f^{\prime }(u)$ = outer stretch factor,
• $\frac{du}{dx}$ = $g^{\prime }(x)$ = inner stretch factor,
• $\frac{dy}{dx}$ = $h^{\prime }(x)$ = total stretch factor.

9. Prerequisite map

Everything flows downhill into the chain rule at the bottom. If any upstream box feels shaky, revisit its section above.

Equipment checklist

Cover the right side and answer aloud — if you can, you're ready for the parent note.