Swap the Two Terms and You Get the Negative of the Answer

You'll learnto run the quotient rule in the one order that works, to square the whole denominator, and to check a quotient derivative against a second, independent route before you trust it.

The product rule adds its two terms, so their order is free — swap them and nothing at all changes. The quotient rule subtracts, and subtraction does not commute. Put the two numerator terms the wrong way round and you do not get a slightly wrong answer; you get the exact negative of the right one, an expression that says the function is climbing at every point where it is actually falling. Everything else here is bookkeeping around that one asymmetry: the numerator's derivative leads, the minus has to reach both parts of what follows it, and the whole denominator — not the top, not the derivative of the bottom — is what gets squared. So this file runs the rule at a point, checks it against a quotient that can be differentiated a second and completely separate way, and then executes the reversal and lets the picture refuse it.

8 STEPS · 6 QUICK CHECKS · u′v LEADS, uv′ FOLLOWS · SWAP THEM AND EVERY SIGN FLIPS · v1

(u/v)′ = (u′v − uv′)/v² ⇄ reverse it and the sign flips

Before you start

What you're looking at

The quotient f(x) = (2x + 1)/(x² + 1). Its denominator is never zero, so there is no asymptote anywhere and the whole curve fits in one window.

The question

The rule subtracts two terms. What does putting them in the other order actually cost — and how would you catch it if you did?

Watch for

The two products being assembled at every point, two independent routes landing on the same answer, and the moment a reversed order draws a tangent climbing across a curve that is falling.

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