Foundations — Product rule — proof

This page assumes you know nothing. Before we can even read the parent statement $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$, we must earn every mark on that line. We go one symbol at a time, and each one gets a plain meaning, a picture, and a reason the topic needs it.

1. A function — the symbol $u(x)$

The letter $u$ is just a name for the machine (like calling a dog "Rex"). The $x$ in brackets is the number you put in. So $u(x)=x^2$ means "the machine that squares whatever you give it": feed in $3$, get $9$.

Why the topic needs it. The parent note talks about a rectangle whose width changes as $x$ changes. "A width that depends on $x$" is exactly a function — we call it $u(x)$. The height is another function, $v(x)$.

Look at the picture: as the input $x$ slides along the bottom, the machine $u$ produces a height on the vertical axis. That curve is the function.

2. A tiny nudge — the symbol $h$

We care about $u(x+h)$: the machine's output after we nudge the input by $h$. If $h$ is small, $x+h$ is close to $x$, so $u(x+h)$ is close to $u(x)$ — but not exactly equal, because the machine reacted to the nudge.

Why the topic needs it. The whole proof asks: "when I nudge $x$ by a hair $h$, how much does the area change?" You cannot ask about change without a nudge to change by. That nudge is $h$.

3. How much the output moved — the symbol $\Delta u$

$\Delta u$ is one number: how far the machine's answer jumped when we stepped the input by $h$.

Look at the red vertical gap in the figure — that gap is $\Delta u$. When $h$ is small the gap is small; shrink $h$ toward zero and the gap shrinks too.

Why the topic needs it. In the parent's rectangle, the right strip has area $v\Delta u$ and the top strip has area $u\Delta v$. The whole geometric story is told in $\Delta u$ and $\Delta v$.

4. The rate of change — the derivative $u^{\prime }$

Now the star of the whole chapter.

The little tick mark ${}^{\prime }$ is the whole notation: $u^{\prime }$ means "the derivative of $u$". Two more marks you'll meet:

• $\frac{\Delta u}{h}$ is called a difference quotient — the average steepness across one step.
• As $h$ shrinks, that average becomes the instantaneous steepness.

The blue line through two points is the average slope $\Delta u/h$. As the second point slides toward the first (the yellow arrows), the line tips into the green tangent — the steepness right at $x$. That limiting slope is $u^{\prime }(x)$. This picture is the entire engine of the parent proof; see Limit definition of the derivative for the full build.

Why the topic needs it. The product rule is a statement about derivatives: it tells you $u^{\prime }$ and $v^{\prime }$ combine into $(uv{)}^{\prime }$. Without the derivative there is nothing to prove.

5. The limit — the symbol $\underset{h\to 0}{\lim }$

We can't just set $h=0$ in $\frac{\Delta u}{h}$, because dividing by zero is forbidden. The limit is the honest way to ask "where is this ratio going?" while $h$ is still a hair above zero.

Why the topic needs it. Every derivative in the proof is a limit. And a key move in the proof, ${\lim }_{h\to 0}u(x+h)=u(x)$, is a limit statement that only works because $u$ is continuous — see Differentiability implies continuity.

6. Continuity — "no sudden jumps"

Picture a curve you can draw without lifting your pen. That is continuity. A staircase with a jump is not continuous at the jump.

Why the topic needs it. In the proof, one factor is $u(x+h)$, and we let $h\to 0$ hoping it becomes $u(x)$. That hope is only justified if $u$ is continuous. Luckily, being differentiable forces continuity, so we get it for free — but the parent note rightly insists we name it.

7. Product of two functions — the symbol $uv$

If $u(x)$ is a width and $v(x)$ is a height, then $u(x)v(x)$ is the area of the rectangle they form. That single product is the object the whole topic differentiates.

The figure shows why the area change has four pieces: original block, right strip $v\Delta u$, top strip $u\Delta v$, and the tiny corner $\Delta u\Delta v$. Everything the product rule says lives in this picture.

8. Multiplying out — algebra we lean on

Two small algebra habits the proof uses:

Prerequisite map

Equipment checklist

Connections

• Limit definition of the derivative — where $u^{\prime }$ and the limit come from.
• Differentiability implies continuity — why ${\lim }_{h\to 0}u(x+h)=u(x)$ is allowed.
• Power rule — the first rule these foundations bootstrap.
• Quotient rule — the same machinery for division.
• Chain rule — the same machinery for composition.
• Leibniz rule (nth derivative of a product) — the grand generalisation of the parent.