Picture a grassy hill, and every walking path that winds over it. Now run the whole landscape through a machine that stretches it vertically to three times the height — every point of ground is hoisted to triple its old altitude. What happens to the paths? A stretch that ends three times higher must climb three times as much over the same horizontal distance. Every path, at every point, is now exactly three times as steep. The gentle parts are three times as gentle-ish, the brutal parts three times as brutal — the shape of the terrain's steepness is unchanged, just multiplied through by 3.

That is the whole of today's rule. Multiplying a function by a constant stretches its graph vertically, and a vertical stretch scales every slope on the graph by the same factor. In derivative language: a constant multiplier floats straight through the derivative untouched. You never fight it, never expand it, never differentiate it — you set it politely to one side, differentiate what's left, and multiply it back on at the end.

The power rule handles the bare powers x^n. But functions in the wild almost never arrive bare — they come dressed with coefficients, like 5x^2, -4x^3 or \tfrac{1}{2}x^4. This one small rule is what lets the power rule loose on all of them.

Scaling the curve scales the slope

Here's the hill-stretching machine in action on y = x^2. The faint curve is the original; the bold one is y = c\,x^2. The marked point sits at a fixed x, so the only thing changing as you drag is the stretch factor c — and the tangent line's slope reads c \cdot 2x, faithfully tracking it.

Watch two things as you slide. At c = 1 the bold curve lies on top of the faint one and the slope is the plain 2x of the power rule. Push c up and the curve rears — the rise over any little run gets multiplied by c, while the run itself doesn't change at all. Slope is rise over run, so slope picks up exactly one factor of c. Not c^2, not c + \text{something} — just c, once.

That's why doubling a function doubles its derivative, halving it halves the derivative, and multiplying by a negative constant flips every slope's sign: the graph is reflected as it's stretched, so uphill becomes downhill by the same amount.

Why it works: the constant escapes the limit

A picture is persuasive, but the difference quotient makes it airtight — and the proof is two lines long. Let's do it concretely for g(x) = 5x^2. Build the difference quotient and watch where the 5 sits:

Both terms in the numerator carry the same factor of 5, so it factors clean out of the top — and then clean out of the whole fraction. What remains inside is exactly the difference quotient of the plain x^2. A constant factor also passes freely through a limit (five times a quantity closing in on L closes in on 5L), so:

Nothing in that argument used the 5 being a 5, or the function being x^2. Replace them with any constant c and any differentiable f and the same two moves — factor c out of the quotient, carry it out of the limit — give the rule in full generality:

This is worth doing from the definition once in your life — after today you'll simply write the answer down. But knowing the constant genuinely escapes the limit, rather than taking it on faith, is what separates using a rule from understanding one.

Read it off the graph

Here is the rule as a double portrait. The bold parabola is f(x) = 3x^2; the faint one is the un-scaled x^2. The bold dashed line is the derivative f'(x) = 6x; the faint dashed line is the un-scaled 2x.

Pick any x and read vertically. At x = 1: the plain parabola sits at height 1 with slope 2; the bold one sits at height 3 with slope 6. At x = 2: heights 4 and 12, slopes 4 and 12. Everywhere you look, the bold curve is exactly three times as tall and exactly three times as steep — the derivative line 6x is just 2x put through the same three-fold stretch. One constant, scaling both stories at once.

Teamwork with the power rule

In practice you'll use this rule welded to the power rule as a single reflex. To differentiate c\,x^n: keep the c, power-rule the x^n, and multiply the two numbers out front together:

Example 1. \dfrac{d}{dx}\bigl[7x^4\bigr]. Hold the 7; the power rule turns x^4 into 4x^3; multiply back:

Example 2 — a negative constant. \dfrac{d}{dx}\bigl[-4x^3\bigr]. The minus sign is part of the constant c = -4, and it rides along like any other constant:

Geometrically that minus sign makes sense: -4x^3 is 4x^3 flipped upside down, so wherever the original climbed, the flipped copy descends at four times x^3's rate — every slope negated and quadrupled.

Example 3 — a fractional constant. \dfrac{d}{dx}\bigl[\tfrac{1}{2}x^4\bigr] = \tfrac{1}{2}\cdot 4x^3 = 2x^3. Fractions squash instead of stretch, but the arithmetic is identical: the two front numbers just multiply. A few more, at a glance:

Notice the last two rows. A constant needn't be a tidy integer — \pi is just a number, so it floats through like any other. And a lone minus sign is the constant c = -1 in disguise: differentiating -f(x) always just negates f'(x).