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

You'll learnthe quotient rule in the one order that works, why the whole denominator is squared, and how one glance at the graph catches a reversed sign.

The product rule adds its two terms, so their order is free. The quotient rule subtracts, and subtraction does not commute: put the two terms the wrong way round and you get the exact negative of the answer. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · u′v LEADS, uv′ FOLLOWS · SWAP THEM AND EVERY SIGN FLIPS · v2

(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 the whole curve fits in one window.

The question

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

Step 1 / 12