A polynomial is a sum of power terms — things like 5x^4, 3x^2, 2x and a constant. You already have every tool you need to differentiate one, because a polynomial is built from exactly the pieces you've learned to handle.

Three rules combine to do the whole job:

• the power rule turns x^n into n x^{n-1};
• the constant multiple rule lets a number out front ride along;
• the sum and difference rule lets you work one term at a time.

The recipe

Put the three rules together and a single mechanical recipe falls out. For each term c\,x^n:

Multiply by the exponent, then drop the exponent by one — and keep the coefficient along for the ride. Do that to every term, carry the +/- signs through, and you're done. A lone constant term has slope 0, so it just disappears.

A 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.

See the slope curve

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

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:

And here is Sal working a full example: