Foundations — Quotient rule — proof

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The parent proof throws a lot of notation at you very fast: $u(x)$, $v(x)$, $f^{\prime }(x)$, ${\lim }_{h\to 0}$, $h$, difference quotients, "continuous", $v^2$. If any of those are fuzzy, the proof will feel like symbol-pushing. Below we earn each symbol one at a time, in the order the proof uses them, with a picture for each.

1. A function: a machine that takes a number and returns a number

The picture. Think of a vending machine: press button $x$, receive snack $u(x)$. The graph draws every (input, output) pair as a point $(x,u(x))$, tracing a curve.

Cloze check: the notation $v(x)$ means the ==output of the machine $v$ when the input is $x$==.

2. The letter $h$: a tiny nudge to the input

The picture. Stand at $x$ on the horizontal axis. Take a small step of length $h$ to the right; you land at $x+h$. The output changes from $u(x)$ to $u(x+h)$.

3. The difference quotient: average steepness over one step

Why a ratio, and why this ratio? Steepness is not "how much did output change" alone — climbing $2$ units is gentle over a long run and steep over a tiny run. Steepness is output-change per unit of input-change. Division answers exactly the question "per unit," so we divide.

4. The limit ${\lim }_{h\to 0}$: what the ratio settles toward

The picture. As $h$ shrinks, the two points on the graph slide together. The straight line through them tilts until it becomes the tangent line — the line that just grazes the curve at $x$. Its slope is the value the difference quotient approaches.

See Limit definition of derivative for the full engine.

5. The derivative $u^{\prime }(x)$: instantaneous rate of change

• $u^{\prime }$ is again a function: give it an $x$, it returns the slope there.
• $u^{\prime }(x)>0$ means the curve is rising at $x$; $u^{\prime }(x)<0$ means falling; $u^{\prime }(x)=0$ means flat.

6. The fraction $\frac{u}{v}$ and its denominator condition $v\ne 0$

All cases of the sign of $v$. The rule works whether $v(x)$ is positive or negative — the only forbidden value is exactly $0$. Because we divide by $v^2$ (see §8), and a square is never negative, the shape of the formula is the same on both sides; only the value of $v$ inside changes.

7. Continuity: the graph has no gaps or jumps

The picture. A continuous curve is one unbroken stroke. A discontinuous one has a break — at the break, nudging the input a hair can jump the output far away, so $v(x+h)$ would not approach $v(x)$.

8. Powers and the notation $v^2$

Prerequisite map

Equipment checklist

When every reveal above is instant, you are ready for Quotient rule — proof.