Drop a ball and the distance it has fallen grows with the square of the time; a widening ripple's area, an inflating balloon's volume, a savings pot left to grow all follow powers too. To find how fast any of them is changing you need the slope of a power — and this page turns that into a one-step rule.

You have earned this page. Four times now you've fed a function into the difference quotient, expanded, cancelled the stubborn h, and taken a limit — once for a constant, once for x, once for x^2, once for x^3. Each answer took a page of algebra. Now line the four answers up side by side, and watch them confess:

Look at what happens to the exponent as you climb the ladder. Reading x^0, x^1, x^2, x^3, each derivative brings the old power down to the front as a multiplier and drops the power by one. Once for a square (2x^1), once for a cube (3x^2) — and if you dared to guess the next rung, you'd write x^4 \to 4x^3. You'd be right. The rung after that? 5x^4. Right again. The pattern you discovered one power at a time holds for every power, and today it finally gets its name: the power rule.

You can watch the degree fall. Here is y = x^3 with its first three derivatives — a cubic (y), then a parabola (y'), then a line (y''), then a flat constant (y'''). Each curve sits one whole degree lower than the one before it, exactly as "drop the power by one" predicts. Those prime marks are the prime notation you met earlier — shorthand for "the derivative, then its derivative, and so on".

And, just for fun, differentiation flattens more than curves. Here's what four derivatives do to a smile:

\tfrac{d}{dx} 🙂 = 🫤  \tfrac{d}{dx} 🫤 = 😐  \tfrac{d}{dx} 😐 = 😮  \tfrac{d}{dx} 😮 = 😵

The pattern, stated once and for all

Here it is — the single most-used fact in all of differential calculus:

Check it against the ladder you built by hand: x^3 \to 3x^2 (✓), x^2 \to 2x^1 = 2x (✓), x^1 \to 1\cdot x^0 = 1 (✓), and even x^0 \to 0\cdot x^{-1} = 0 (✓) — the constant rule falls right out as the bottom rung. Four hard-won theorems have just become four easy special cases of one line.

Appreciate what you're being handed here. Every derivative so far cost you a difference quotient, an expansion, a cancellation and a limit. From this point on, for any power of x, the whole ritual collapses into a two-second stencil: power down, power drops. The limits haven't vanished — they're baked in, done once in general so you never have to do them again. That's what a rule is: a limit argument, pre-paid.

Worked examples: no limits required

Example 1 — f(x) = x^5. The old way would demand expanding (x+h)^5 — six terms, five minutes, three chances to drop a coefficient. The new way: the power is 5, so bring the 5 down and knock the exponent to 4:

Done. Example 2 — g(x) = x^{10}. Power down, power drops:

Example 3 — evaluate a slope. How steep is y = x^5 at x = 2? Differentiate first, substitute second: y' = 5x^4, so the slope there is 5 \cdot 2^4 = 5 \cdot 16 = 80. The tangent at (2, 32) rises eighty units for every one across — that's how savagely a fifth power grows.

And the party trick: \tfrac{d}{dx}[x^{100}] = 100x^{99}, written faster than you can say "difference quotient". A century of powers, one second of work.

Try every power at once

A rule that claims to work for every n deserves to be stress-tested. Slide n and watch the curve x^n (solid) next to its derivative n\,x^{n-1} (dashed). Two things to hunt for as you slide: the dashed curve always sits one degree lower than the solid one — a quartic's derivative is a cubic, a cubic's is a parabola — and wherever the solid curve is momentarily flat, the dashed curve crosses zero at exactly that x. The derivative isn't just a formula that resembles the function; it's a running commentary on the function's slope, point by point.

Try n = 2 versus n = 3 and look at the dashed curves. For x^2 the derivative 2x is negative on the left — the parabola runs downhill there. For x^3 the derivative 3x^2 never dips below zero — a cubic climbs on both sides, pausing flat only for an instant at the origin.

The derivative is the slope

One more sanity check before we push the rule somewhere new. Here's y = x^3 (solid) with the straight tangent line that just grazes it (dashed). The power rule promises the slope at x = a is 3a^2. Slide a along the curve and test the promise: at a = 1 the tangent should climb with slope 3; at a = 2 a steep 12; at a = -1, slope 3 again — because 3a^2 can't be negative, the cubic leans uphill on both sides. And at a = 0 the tangent lies perfectly flat along the axis.

Why it works: the binomial theorem's opening move

A pattern spotted four times isn't a proof — maybe the fifth power breaks it? Here's the sketch that settles every power at once, and it's just the general version of the cube picture from the x^3 page. Expand (x + h)^n with the binomial expansion. You only need to look at its first two terms:

Read it geometrically, the way you read the cube: an n-dimensional "cube" of side x grows by a whisker h. The new volume is the old x^n, plus n thin slabs — one glued to each of the n faces, each of size x^{n-1} h — plus crumbs in the corners, all of them carrying h^2 or worse. Now run the difference quotient:

The x^ns cancel, one h cancels, and everything left over still holds at least one factor of h — so as h \to 0, the crumbs melt away and only the slabs survive:

That's where both halves of the stencil come from. The n out front counts the n faces of the growing cube; the x^{n-1} is the area of each face. The rule isn't a coincidence you memorise — it's geometry you can picture.

The twist: it isn't just for whole numbers

Here's the part nobody would dare guess from the ladder. The rule keeps working when n is negative or a fraction — you just have to rewrite the function as a power first. This rewrite-then-differentiate move is where most exam marks in this topic live, so watch both classics in slow motion.

Rewrite 1 — reciprocals are negative powers. Since \tfrac{1}{x^2} = x^{-2}, the power rule applies with n = -2: bring the -2 down, then drop the exponent by one — from -2 down to -3:

The minus sign is not an accident — it's information. 1/x^2 is a falling function (for x > 0, bigger input means smaller output), so its slope must come out negative. The simplest case works the same way: \tfrac{1}{x} = x^{-1} \Rightarrow -1 \cdot x^{-2} = -\tfrac{1}{x^2}.

Rewrite 2 — roots are fractional powers. Since \sqrt{x} = x^{1/2}, apply the rule with n = \tfrac12. Down comes the half; the exponent drops from \tfrac12 to \tfrac12 - 1 = -\tfrac12:

Same recipe every time: rewrite as a power, power to the front, drop it by one — and then, usually, rewrite back into root-or-fraction form for the final answer. Here's the square root with its derivative. Watch how \tfrac{1}{2\sqrt{x}} behaves at the two ends: near x = 0 it blows up — the root curve launches almost vertically — while far to the right it sinks toward zero, because \sqrt{x} keeps rising but ever more lazily.

A small dictionary of disguised powers worth carrying into any exam: \tfrac{1}{x} = x^{-1}, \tfrac{1}{x^3} = x^{-3}, \sqrt{x} = x^{1/2}, \sqrt[3]{x} = x^{1/3}, \tfrac{1}{\sqrt{x}} = x^{-1/2}, and x\sqrt{x} = x^{3/2}. Translate first; differentiate second.