Saturday, April 10, 2010

Another fun proof

I've posted here repeatedly about how much I like Strang's Calculus. Here is another example of his incisive and insightful style.

The product rule is a very useful result from differential calculus:

(uv)' = u v' + v u'

A second rule that is somewhat harder to remember applies to quotients:

(u/v)' = (v u' - u v')/v2

Strang has a beautiful proof of the quotient rule.
He starts by deriving a third one---the reciprocal rule. Start with this identity:

v (1/v) = 1

Use the product rule:

d/dx [ v (1/v) ] = 0
v (1/v)' + (1/v) v' = 0
(1/v)' = -v'/v2

Now use the reciprocal rule and the product rule to derive the quotient rule:

d/dx(u/v) = (1/v) du/dx + u d/dx(1/v)
(u/v)' = (1/v) u' - u (1/v)'
(u/v)' = (1/v) u' - u v'/v2
= (v u' - u v')/v2

Very pretty!

UPDATE: David Jerison uses the chain rule and the power rule to derive the reciprocal rule:

(1/v)' = (v-1)' = -v-2 v'

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