Is the derivative of a product the product of the derivatives?
Updated
The short answer
The derivative of f(x)g(x) is f′(x)g(x) + f(x)g′(x), so each factor takes a turn being differentiated while the other stays the same. For x² sin x that gives 2x sin x + x² cos x, and the usual slip is multiplying the two derivatives to get 2x cos x.
What it looks like
| Function | Wrong derivative | Right derivative |
|---|---|---|
| x² sin x | 2x cos x | 2x sin x + x² cos x |
| x³ · x⁶ | 3x² · 6x⁵ = 18x⁷ | 3x² · x⁶ + x³ · 6x⁵ = 9x⁸ |
The second row is a check you can run on any product you can simplify first, and it's the one Paul Dawkins uses. Since x³ · x⁶ is just x⁹, its derivative is 9x⁸. Multiplying the two derivatives gives 18x⁷, which doesn't match. The product rule gives 3x⁸ + 6x⁸ = 9x⁸, which does.
In alai, an iPad app that reads your handwriting as you work, you can hold to talk and ask the tutor, Ally, why a step works.
Why it happens
The derivative of a sum really is the sum of the derivatives, so it feels as if products should work the same way. They don't, because when two changing things are multiplied, the product changes for two reasons at once.
Picture a rectangle with sides f and g, so its area is fg. When x moves a little, both sides grow a little, and the area gains two strips. One runs along the side of length g and comes from f growing. The other runs along the side of length f and comes from g growing. Those strips give the two terms, f′g and fg′. There's also a tiny corner where the two growths meet, and it's so small that it drops out of the derivative. The wrong answer, f′g′, describes only that corner.
How to catch it yourself
Count the terms. The derivative of a product of two factors has two terms, and in each one, one factor is differentiated while the other is copied as it is. Then point at the sin x in the problem and ask yourself:
When you differentiated the x², what happened to the sin x?
A tiny example also settles it. x · x = x², whose derivative is 2x. Multiplying the derivatives gives 1 · 1 = 1, while the product rule gives 1 · x + x · 1 = 2x.
Try these
Differentiate each one.
- x sin x
- x² eˣ
- (x + 1)(x² − 3)
Answers
- sin x + x cos x
- 2x eˣ + x² eˣ
- 3x² + 2x − 3, which you can check by multiplying out first: (x + 1)(x² − 3) = x³ + x² − 3x − 3
How alai helps
If you want to know why the product rule has two terms, hold to talk and ask, and Ally answers out loud with a short reply. Ally also remembers what tripped you up in past sessions, so a later session can start from there.