Pure · Differentiation
Chapter 1 · 4
The idea
The product rule
Differentiating a product of two functions, y = uv, as u'v + uv' — why the naive "differentiate each and multiply" is wrong, and how to combine the rule with the chain rule.
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Pure · Differentiation
The product rule
Differentiating a product of two functions, y = uv, as u'v + uv' — why the naive "differentiate each and multiply" is wrong, and how to combine the rule with the chain rule.
Why it works
Why the naive answer is wrong
When two functions are multiplied — say — the temptation is to differentiate each factor and multiply the results. That is wrong, and it is worth seeing why before learning the fix.The growing rectangle
A product measures an area. Picture a rectangle whose width is and height is , so its area is . Nudge a little: the width grows by a sliver and the height by a sliver . The area grows by three pieces — a tall thin strip down the side, a long thin strip along the top, and a tiny corner rectangle . That corner is the product of two small things, so it is vanishingly small compared with the strips and disappears in the limit. The change in area is thereforeKeep reading — free
The rest of the explanation, plus 3 worked examples you step through move by move.
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