Foundations — Sum, difference, constant multiple rules

Before you can trust the sum, difference, and constant-multiple rules, you need to be fluent in every symbol they lean on. This page builds each one from nothing — plain words, then a picture, then why the topic needs it. Read top to bottom: each block uses only what came before.

1. A function $f(x)$ — a machine that turns numbers into numbers

The picture. Think of a vending machine. You press a button ($x$), you get one snack ($f(x)$). Same button always gives the same snack — that "exactly one output" part matters.

Why the topic needs it. The parent note keeps writing things like $f(x)+g(x)$ or $3x^4-5x^2+7$. Those are just functions. You cannot talk about "the slope of a curve" until you agree the curve is a function's output drawn as a graph.

2. Slope — how steep a straight line is

The picture. Pick two points on a line. Walk right by some amount (the run). Count how far up (or down) you climbed (the rise). Divide. A steep hill has big rise for small run — big slope. A flat road has zero rise — zero slope. Downhill gives a negative slope.

Why the topic needs it. "Derivative" is going to mean slope of a curve. But a curve bends, so its steepness changes from point to point. We first nail slope for straight lines, then extend it.

3. The symbols $\Delta x$ and "change in" — measuring a step

The picture. Stand at input $x$. Take a tiny step right of size $\Delta x$, landing at $x+\Delta x$. The output jumps from $f(x)$ up (or down) to $f(x+\Delta x)$. The vertical jump is the rise; the step $\Delta x$ is the run.

Why the topic needs it. This is exactly rise-over-run, but for a curve, between two nearby points: $\frac{f(x+\Delta x)-f(x)}{\Delta x}=\frac{\text{rise}}{\text{run}}$ This fraction is called the difference quotient — memorise its shape, the whole derivative is built on it. See Definition of the derivative (limit of difference quotient).

4. The limit $\lim$ — "what does it approach?"

The picture. Shrink your sideways step $\Delta x$ smaller and smaller. The two points on the curve slide together. The rise-over-run line stops being a "chord across two points" and becomes the tangent — the line that just grazes the curve at one spot. Its slope is the number the difference quotient approaches.

Why this tool and not just plugging in $0$? Because the difference quotient literally divides by $\Delta x$. Setting $\Delta x=0$ divides by zero. The limit is the only honest way to ask "what slope are we approaching as the two points merge?" without ever dividing by zero.

5. The derivative $f^{\prime }(x)$ and $\frac{d}{dx}$ — slope of the curve at a point

Why the topic needs both. The parent proofs start from $f^{\prime }(x)=\lim \dots$ (Lagrange, compact) but state the rules as $\frac{d}{dx}[f+g]$ (Leibniz, shows which variable). You must recognise they are the same object wearing two costumes.

6. A constant $c$ and its zero slope

The picture. A flat line never rises. Rise $=0$ for any run, so slope $=0$. That is why $\frac{d}{dx}(7)=0$: a flat road has no steepness.

Why the topic needs it. The constant-multiple rule $\frac{d}{dx}[cf]=c{f}^{\prime }$ needs $c$ to be a genuine constant so it can "ride out" of the limit untouched. And the "derivative of a constant is $0$" fact is exactly why the $+7$ in Example 1 vanishes.

7. Building functions: sum, difference, scalar multiple

The picture. To read $f+g$ at some $x$: find $f$'s height, find $g$'s height, stack one on top of the other. The new curve's height is the sum. Scaling by $c$ multiplies every height by $c$.

Why the topic needs it. These are the only three operations linearity covers. The parent's whole message — "adding stays added, scaling stays scaled" — is a statement about exactly these three moves and no others (notably not multiplication of two functions; that needs the Product rule).

8. Limit laws — why we may split a limit apart

The picture. If two quantities each settle down to a value, then their running total settles to the total of those values — the wobbles can't fight because each dies down on its own. Scaling everything by $c$ just scales the destination by $c$.

Why the topic needs it. These laws are the engine of every proof in the parent. When the sum-rule proof splits one big difference quotient into two, it is the sum law that licenses the split; when $c$ escapes the limit in the constant-multiple proof, it is "constant pulls out". Full details live in Limit laws (sum, scalar, product of limits).

9. Linearity — the one word that names the whole topic

Why the topic needs it. The parent's grand conclusion is that $\frac{d}{dx}$ is a linear operator — differentiation is a linear operation. The sum rule, difference rule, and constant-multiple rule are the three visible faces of this single fact. See Linear operators for the abstract view and Linearity of integration for its integral twin.

Prerequisite map

Everything on the left feeds the derivative; the derivative plus the combining moves plus the limit laws give linearity — the parent topic.

Equipment checklist

Cover the right side and test yourself. If any answer is shaky, re-read that section before opening the parent note.