The product rule, visually

What you’ll learn

Functions rarely arrive alone; the most basic way they combine is multiplication. The single capability this lesson builds: apply the product rule and explain it as the changing area of a rectangle, so you understand where its two terms come from instead of memorizing a formula (and stop reaching for the wrong one-term guess).

The tempting first guess, that the derivative of f · g is f' · g', is wrong; the correct product rule is d/dx(f · g) = f' · g + f · g', a sum of two terms. One picture explains it: let f be a rectangle’s width and g its height, so f · g is the area. Nudge x by dx and the area gains a top strip (f · g' · dx), a side strip (f' · g · dx), and a tiny corner block (f' · g' · dx²). Divide by dx and let it shrink: the two strips survive (the two terms), the corner vanishes, and the corner is exactly the wrong f' · g' guess. You will work several examples (x² · x³, cross-checked against the power rule; x · sin x; sin x · cos x), see a numeric nudge confirm the corner is negligible, and extend the rule to three or more factors (one term per factor).

Where this fits

This is lesson 5 of Phase 2 (The differentiation toolkit) and its opener. Phase 1 built the derivative and the rules for single functions (powers, trig); Phase 2 is about combining functions. This lesson handles products; the next (lesson 6) handles the other combination, nesting, with the chain rule, and the two come from the same 3B1B chapter and the same rectangle/nudge reasoning. Together they let you differentiate almost any combination you will meet. The examples here lean on the power rule (lesson 3) and trig derivatives (lesson 4), so the rules start cooperating.

Before you start

Prerequisite (within this track): lesson 4, Trig derivatives from geometry, since the worked examples use cos x and sin x derivatives, and the nudge-and-look method carries straight over. You also want the power rule (lesson 3) fresh, since one example cross-checks against it. Comfort multiplying simple expressions is all the algebra needed; no coding, nothing installed. The practice is pen and paper.

By the end, you’ll be able to

• Apply the product rule d/dx(f*g) = f’g + fg’ to products of functions
• Explain the product rule as the changing area of a rectangle (a top strip plus a side strip), and why the corner block vanishes
• Explain why f’*g’ is wrong (it is the vanishing corner) and why the rule has exactly two terms
• Extend the product rule to three or more factors, one term per factor

Time and difficulty

• Read time: about 11 minutes
• Practice time: about 13 minutes (applying the product rule with a power-rule cross-check, a numeric corner-vanishes check, and flashcards)
• Difficulty: standard