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Pure · Differentiation

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Differentiating powers of x

Bring the power down to the front and knock it down by one — and why that rule is exactly what the gradient-of-a-chord limit always produces.

Pure · Differentiation

Differentiating powers of x

Bring the power down to the front and knock it down by one — and why that rule is exactly what the gradient-of-a-chord limit always produces.

Why it works

Differentiation has one job: find the gradient of a curve — how steep it is — at a point. The catch is that "gradient" was only ever defined for straight lines (rise over run). A curve's steepness changes everywhere, so what does its gradient at a single point even mean?

The trick is to sneak up on it. Take two points on the curve and join them with a straight line — a chord — whose gradient you can find, it's just rise over run. Now slide the second point towards the first. The chord pivots, and its gradient closes in on the steepness of the curve at that one point: the tangent. That limit is the derivative.

Watch it fall out for y=x2y = x^2. Step from xx to a hair further along, x+hx + h. The rise is (x+h)2−x2=2xh+h2(x+h)^2 - x^2 = 2xh + h^2; the run is hh. So the chord's gradient is 2xh+h2h=2x+h.\frac{2xh + h^2}{h} = 2x + h. Now let hh shrink to nothing: the hh on the end disappears and you're left with 2x2x. So the gradient of y=x2y = x^2 is 2x2x — and look what happened to the power: the 22 came down to the front, and the power dropped from 22 to 11.

At x=1x = 1 that says the gradient is 22 — and you can see it: the tangent to the curve at (1,1)(1, 1) climbs 22 up for every 11 across.-3-2-11232468(1, 1)xyDo the same with x3x^3 and you get 3x23x^2; with x4x^4, 4x34x^3. The pattern never changes, and it is the rule: ddx xn=n xn−1.\frac{d}{dx}\,x^n = n\,x^{n-1}. Bring the power down to the front, then knock the power down by one. It's not a magic spell — it's what the chord-gradient limit always spits out.

A few things drop straight out of this:
  • A constant has gradient 00. The line y=7y = 7 is flat — no steepness — so
ddx(7)=0\frac{d}{dx}(7) = 0. (In the rule, 7=7x07 = 7x^0; the power 00 comes to the front and kills the term.)
  • xx on its own differentiates to 11, since x=x1→1⋅x0=1x = x^1 \to 1\cdot x^0 = 1.
  • Constant multiples just ride along: ddx(5x3)=5⋅3x2=15x2\frac{d}{dx}(5x^3) = 5\cdot 3x^2 = 15x^2,
and you differentiate a sum one term at a time.

One practical catch — exactly as in integration: the rule only speaks powers of xx. So x\sqrt{x}, 1x2\frac{1}{x^2} and friends must be rewritten as x1/2x^{1/2}, x−2x^{-2} before the rule can touch them. Once you're comfortable there, negative and fractional powers stop being scary: "knock the power down by one" still just means subtract one, even when that turns −2-2 into −3-3.