The arc of a thrown ball, a roller-coaster's track, a company's profit as it changes its prices — real curved formulas are everywhere, and a constant question about them is "how steep is it here, and where does it level off?" Most of those curves are polynomials, and this page hands you a fast recipe to differentiate any of them.

This is the toolbox moment. Over the last few lessons you collected three small rules, each proved on its own, each modest by itself: the power rule turns x^n into n x^{n-1}; the constant multiple rule lets a number out front ride along unchanged; and the sum and difference rule lets you take a sum apart and work one piece at a time.

Snap the three together and something remarkable happens: you can now differentiate any polynomial whatsoever. Not just x^2, not just tidy textbook cubics — any sum of power terms, however long, whatever its coefficients. A polynomial is exactly a sum of pieces like 5x^4, -3x^2, 2x and a lone constant — and every one of those pieces is something your three rules already handle. There is nothing in a polynomial the toolbox can't reach.

Feel the scale of the shortcut. When Newton wanted the slope of a curve, he had to run a limit argument: form the difference quotient, expand (x+h)^n, cancel, and let h \to 0 — a careful page of algebra every single time. You did that work once, per rule, and banked the results. Now a polynomial of degree fifty takes you ten seconds, term by term, no limits in sight. That's what a good theorem is: a limit argument you never have to run again.

The recipe

Put the three rules together and a single mechanical recipe falls out. Every term of a polynomial has the shape c\,x^n — a coefficient times a power — and for each such term:

Multiply by the exponent, then drop the exponent by one — and keep the coefficient along for the ride. The sum and difference rule says you may do this to every term independently, carrying the +/- signs through untouched. Two special cases are worth saying out loud, because they're where slips happen:

• a term like 7x is really 7x^1, so it becomes 7 \cdot 1 \cdot x^0 = 7 — an x-term turns into a plain constant;
• a lone constant like -9 is a horizontal line with slope 0 everywhere, so it simply vanishes.

Note what the theorem quietly promises: differentiable everywhere. Polynomials have no corners, no jumps, no vertical tangents — they are among the smoothest functions in all of mathematics, and their derivatives are polynomials again, so you can differentiate a second time, a third time, as often as you please.

In fact, try it: differentiate x^4 over and over and watch it step down the staircase,

the degree ticking down by one at every pass until a constant falls off the end and everything after it is 0 forever. A degree-n polynomial survives exactly n differentiations before flatlining — a fact you'll lean on when second derivatives arrive.

A first worked example

Differentiate p(x) = 2x^3 - 4x^2 + 5x - 7. Take it one term at a time:

• \dfrac{d}{dx}\big[2x^3\big] = 2\cdot 3\,x^2 = 6x^2
• \dfrac{d}{dx}\big[-4x^2\big] = -4\cdot 2\,x = -8x
• \dfrac{d}{dx}\big[5x\big] = 5
• \dfrac{d}{dx}\big[-7\big] = 0

Reassemble:

Notice the derivative of a degree-3 polynomial is degree 2: each term loses one power, so differentiating always lowers the degree by one. That's your fastest sanity check — before you check any coefficient, check the shape of your answer.

The full assembly, rules cited by name

Once more, slowly — this time naming which rule earns its keep at each move. Differentiate

Move 1 — sum and difference rule. The derivative of a sum is the sum of the derivatives, so we may split the problem into four independent one-term problems and carry the signs through:

Move 2 — constant multiple rule. Each coefficient slides outside its derivative and waits:

Move 3 — power rule, four times over: \frac{d}{dx}[x^5] = 5x^4, \frac{d}{dx}[x^3] = 3x^2, \frac{d}{dx}[x] = 1, and the constant 9 differentiates to 0. Substitute and tidy:

Degree 5 in, degree 4 out — the shape checks. With practice you'll stop writing the middle lines and jump straight from q to q': the three rules fuse into one reflex, multiply down, knock the power down, constants die. But it's worth having seen, at least once, exactly which theorem justifies each stroke of the pen.

Physics loves this: position, velocity, acceleration

Here's why physicists adore polynomial differentiation: it converts a position formula into a velocity formula, and then — because the derivative of a polynomial is another polynomial — a second pass converts velocity into acceleration. Suppose a drone's height above its launch pad, in metres after t seconds, is

First pass — velocity. Velocity is the rate of change of position, so differentiate term by term (power rule on each power, coefficients riding along, signs carried through):

Second pass — acceleration. Acceleration is the rate of change of velocity. But v(t) is itself just a polynomial, so run the very same recipe again:

Now read the story off the formulas. At launch (t = 0) the drone moves at v(0) = 1 m/s but its acceleration is a(0) = -4 m/s² — it starts off actually losing speed. By t = 2 seconds it's doing v(2) = 60 - 8 + 1 = 53 m/s and accelerating at a(2) = 56 m/s². Two passes of a ten-second recipe, and a cubic position law has surrendered its entire motion story. Notice the degrees stepping down — cubic position, quadratic velocity, linear acceleration — exactly as the theorem promised.

See the slope curve — and hunt the flat spots

The derivative p'(x) is itself a function — it reports the slope of p at every point. Plot them together for p(x) = x^3 - 3x, whose derivative the recipe delivers instantly: p'(x) = 3x^2 - 3. Where the bold curve is flat (its peak and valley), the dashed slope curve crosses zero. Where p climbs steeply, p' is high; where it falls, p' is negative.

And here is the real power move: you can find those flat spots exactly, by algebra, without squinting at any graph. A flat spot is a place where the slope equals zero, so set the derivative to zero and solve:

Check the chart: the peak sits at x = -1 (where p(-1) = 2) and the valley at x = 1 (where p(1) = -2) — precisely where the dashed curve cuts the axis. Nothing limits you to a target of zero, either. Where does the curve have slope exactly 9? Solve 3x^2 - 3 = 9, so x^2 = 4 and x = \pm 2. Differentiate once, and every "where is the slope equal to…?" question collapses into ordinary equation-solving — this is the engine behind finding maximum and minimum values, a whole topic that's waiting for you just ahead.

Watch it built up

Differentiating term by term is so regular it's almost a chant: bring the power down, drop it by one. Watch the rule sweep across a polynomial, term by term — the same first example you worked above, now in motion. Notice how each term is handled in complete isolation: the -4x^2 neither knows nor cares what happened to the 2x^3 before it. That independence is the sum and difference rule doing its job:

And here is Sal working a full example: