Visual walkthrough — Sum, difference, constant multiple rules

Before we touch a single formula, let's agree on what every word means — from zero.

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Step 0 — The one word we must define first: slope

WHAT. A function like $y=f(x)$ is a rule: feed it a number $x$, it hands back a number $y$. Plotting all the pairs $(x,y)$ gives a curve. The slope at a point is the answer to one question: if I nudge $x$ a tiny bit to the right, how fast does $y$ climb? Steep uphill = big positive slope. Downhill = negative slope. Flat = zero slope.

WHY we need it. The symbol $f^{\prime }(x)$ (read "$f$-prime of $x$") is nothing but the slope of the curve $f$ at the point $x$. If you can picture slope, you already understand the derivative — the rest is measuring it precisely.

PICTURE. The red line just grazes the curve at one point; its steepness is $f^{\prime }(x)$.

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Step 1 — Build a sum, and watch the heights stack

WHAT. Take two curves, $f$ (teal) and $g$ (plum). Define a new curve by adding their heights at every $x$: $h(x)=f(x)+g(x)$

WHY. This is the whole object of interest. We want the slope of $h$ — but $h$ was manufactured by stacking two known curves. The question is whether that manufacturing process carries over to slopes.

PICTURE. At the chosen $x$, the burnt-orange height of $h$ is literally the teal segment placed on top of the plum segment. Two bricks stacked; total height = sum of the two.

$h(x)=\underset{\text{teal brick}}{\underbrace{f(x)}}+\underset{\text{plum brick}}{\underbrace{g(x)}}\Rightarrow \text{orange height}=\text{teal}+\text{plum}$

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Step 2 — Nudge $x$: both bricks grow, so the sum grows by the two growths

WHAT. Step right by $\Delta x$. The teal brick grows by an amount we'll call $\Delta f=f(x+\Delta x)-f(x)$. The plum brick grows by $\Delta g=g(x+\Delta x)-g(x)$. Because $h$ is the two stacked, its growth is both growths added: $\Delta h=\Delta f+\Delta g$

WHY. This is the geometric heart of the sum rule. Nothing about the two bricks interacts — one sits on the other, so their increases simply pile up. No products, no cross-terms.

PICTURE. Two shaded growth-strips: a teal strip of height $\Delta f$ and a plum strip of height $\Delta g$. The orange total-growth bar is exactly as tall as the two strips combined.

$\underset{\Delta h,\text{orange bar}}{\underbrace{h(x+\Delta x)-h(x)}}=\underset{\Delta f,\text{teal strip}}{\underbrace{f(x+\Delta x)-f(x)}}+\underset{\Delta g,\text{plum strip}}{\underbrace{g(x+\Delta x)-g(x)}}$

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Step 3 — Divide by the step: two slopes appear side by side

WHAT. Divide everything in Step 2 by the step width $\Delta x$: $\frac{\Delta h}{\Delta x}=\frac{\Delta f}{\Delta x}+\frac{\Delta g}{\Delta x}$

WHY. Rise-over-run turns each raw growth into a slope. We split the fraction because a sum on top of a single denominator can always be shared out term by term — that's just how fractions work: $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$.

PICTURE. Three connecting lines drawn: the orange one for $h$, the teal one for $f$, the plum one for $g$. Their steepnesses satisfy: orange steepness = teal steepness + plum steepness.

$\underset{\text{slope of }h\text{’s little line}}{\underbrace{\frac{\Delta h}{\Delta x}}}=\underset{\text{slope of }f\text{’s line}}{\underbrace{\frac{\Delta f}{\Delta x}}}+\underset{\text{slope of }g\text{’s line}}{\underbrace{\frac{\Delta g}{\Delta x}}}$

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Step 4 — Shrink the step to zero: the slopes become derivatives

WHAT. Let $\Delta x\to 0$. Each little-line slope collapses onto the true grazing-line slope — the derivative: ${\lim }_{\Delta x\to 0}\frac{\Delta h}{\Delta x}={\lim }_{\Delta x\to 0}\frac{\Delta f}{\Delta x}+{\lim }_{\Delta x\to 0}\frac{\Delta g}{\Delta x}$

WHY THIS TOOL — the limit, and this specific limit law. We use a limit because a slope "at a single point" needs the two points to merge; the limit is the only tool that lets $\Delta x$ vanish without dividing by a hard zero. And we're allowed to split the limit across the $+$ sign only because of the sum law for limits: the limit of a sum equals the sum of the limits, provided each limit exists — see Limit laws (sum, scalar, product of limits). Both pieces exist here precisely because we assumed $f$ and $g$ are differentiable.

PICTURE. A fan of connecting lines with $\Delta x$ shrinking $\to 0$; all three fans converge onto their grazing lines simultaneously.

$\boxed{h^{\prime }(x)=f^{\prime }(x)+g^{\prime }(x)}■$

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Step 5 — The constant multiple rule: stretching the curve stretches the slope

WHAT. Now instead of adding, scale. Let $c$ be a fixed number and $h(x)=cf(x)$: multiply every height of $f$ by $c$. Claim: the slope also multiplies by $c$.

WHY. Stretching a curve vertically by $c$ makes every rise $c$ times bigger, while the horizontal run $\Delta x$ is untouched. Rise-over-run therefore scales by $c$. Algebraically, $c$ doesn't depend on $\Delta x$, so it pulls straight out of the limit — the scalar law for limits from Limit laws (sum, scalar, product of limits).

PICTURE. For $c=2$, the plum stretched curve is twice as tall everywhere; its grazing line is twice as steep. For $c=\frac{1}{2}$, half as tall, half as steep.

$h^{\prime }(x)={\lim }_{\Delta x\to 0}\frac{cf(x+\Delta x)-cf(x)}{\Delta x}=c\underset{=f^{\prime }(x)}{\underbrace{\underset{\Delta x\to 0}{\lim }\frac{f(x+\Delta x)-f(x)}{\Delta x}}}=c{f}^{\prime }(x)$

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Step 6 — Difference is not a new rule (and the sign trap it hides)

WHAT. Subtraction is addition of a negative: $f-g=f+(-1)g$. Apply Step 5 with $c=-1$ to $g$ (flip it upside down), then Step 5's sum rule: $\frac{d}{dx}[f-g]=f^{\prime }+(-1)g^{\prime }=f^{\prime }-g^{\prime }$

WHY. No new geometry needed — flipping $g$ upside down flips its slope's sign, then heights (now one negative) still just stack. This is the "difference is free" idea, made visible.

PICTURE. Plum $g$ reflected through the horizontal axis into $-g$; its downhill slope becomes uphill. Stacking teal $+$ flipped-plum gives $f-g$.

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Step 7 — Edge & degenerate cases (never leave a scenario unshown)

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The one-picture summary

One frame: teal $f$ and plum $g$ stacked into orange $h=f+g$, with all three grazing lines drawn, and the little slope-triangle showing orange-tilt = teal-tilt + plum-tilt. That single visual is linearity of the derivative — and its abstract name lives at Linear operators, with the integral twin at Linearity of integration.

Recall Feynman retelling — the whole walkthrough in plain words

Imagine two hills, a teal one and a plum one, and you build a new hill by standing on the teal hill and piling the plum hill's height on top at every point. Now walk one tiny step forward. The teal ground under you rose a bit; the plum pile on top rose a bit more. Your total climb is just those two little climbs added — the piles never fight each other. Divide each climb by your step length and you've got slopes; shrink the step to nothing and slopes become derivatives. So the combined hill's steepness is exactly the two steepnesses added: that's the sum rule. Now if you'd stretched one hill to be twice as tall, every rise doubles while your stride stays the same, so its steepness doubles too — that's the constant multiple rule. And "going downhill on $g$" is the same as flipping $g$ upside down and adding, which is subtraction — no new rule, just a minus sign you must apply to the entire second hill. Adding stays added, scaling stays scaled. That's all.

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Connections

• Sum, difference, constant multiple rules — the parent this page draws.
• Definition of the derivative (limit of difference quotient) — Step 0's slope machine.
• Limit laws (sum, scalar, product of limits) — the sum law (Step 4) and scalar law (Step 5).
• Power rule — pair with linearity to differentiate any polynomial term by term.
• Product rule — the non-linear sibling: heights multiply, and stacking logic breaks.
• Linearity of integration — the same "stacks stay stacked" picture, run in reverse.
• Linear operators — the abstract home of "add stays added, scale stays scaled".