Differentiation Splits a Sum, and Stops at a Product

You'll learnto differentiate a polynomial one term at a time — coefficients riding through untouched, constants vanishing outright — and to name the boundary where that licence runs out.

Add 3 to every output of a function. The whole graph lifts, and not one tangent turns: a vertical shift changes heights, never slopes, so a constant differentiates to 0. Multiply every output by 2 instead and the graph stretches — and so does every slope, by exactly the same factor. Add two functions and their slopes add too. Those three sentences are one property, and the property is called linearity. It turns any polynomial into a single line of work: go term by term, multiply the coefficient by the exponent, drop the exponent by one, keep the sign. What linearity will not do is survive multiplication. Take f(x) = x and g(x) = x. Then (f · g)′ = 2x, while f′ · g′ = 1. Two different functions — so no such rule exists, which is exactly why Topics 2.8 and 2.9 exist.

8 STEPS · 6 QUICK CHECKS · A CONSTANT DIFFERENTIATES TO 0, NOT TO ITSELF · LINEARITY STOPS AT A PRODUCT · v1

f(x) = x³ − 3x² + 3 ⇄ f′(x) = 3x² − 6x

Before you start

What you're looking at

The cubic f(x) = x³ − 3x² + 3, and the four rules that let it be differentiated one term at a time instead of by a limit.

The question

Which parts of an expression does differentiation pass straight through — and which parts does it refuse to?

Watch for

Tangents staying parallel while the whole curve lifts, the same tangents steepening when the curve is stretched, and the moment a surviving constant lifts the entire slope function off its zeros.

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