Archive for September, 2009

On the Product and Quotient Rules

September 27, 2009

I was recently reading in Courant’s Differential and Integral Calculus and I found a much better formulation of the product and quotient rules than what typically appears in modern textbooks (the English edition of Courant’s book, by the way, is now 75 years old.  Does anyone know if this is its last year of copyright in the US?).  The usual formulation of the product and quotient rules is as follows:  Let r(x) = f(x) g(x) and s(x) = \frac{f(x)}{g(x)} where f(x) and g(x) are differentiable functions.  Then:

r^\prime (x) = f(x) g^\prime (x) + f^\prime (x) g(x)

and

s^\prime (x) = \frac{g(x)f^\prime (x) - f(x) g^\prime (x)}{ [g(x)]^2}

Compare this with Courant’s formulation:

\frac {r^\prime (x)}{r(x)} = \frac{f^\prime (x)}{f(x)} + \frac{g^\prime (x)}{ g (x)}

and

\frac {s^\prime (x)}{s(x)} = \frac{f^\prime (x)}{f(x)} - \frac{g^\prime (x)}{ g (x)}

Courant’s formulation of these rules is much simpler (especially in the case of the quotient rule) and show that the product and quotient rules are very closely related (of course they have to be very closely related since division is the inverse of multiplication; what is great about Courant’s formulation is that this relationship is very apparent).

Another point in favor of Courant’s formulation is that is extremely easy to generalize.  It is one easy step from Courant’s formulation to:

\frac{(f_1 f_2 ... f_n)^\prime}{(f_1 f_2 ... f_n)} = \frac{f_1^\prime}{f_1} + \frac{f_2^\prime}{f_2} + ... + \frac{f_n^\prime}{f_n}

This gives a simple proof of the power rule for natural numbers (Courant proves the power rule this way in his book).  It is also just one easy step to full product/quotient rule:   If q = \frac{(f_1 f_2 ... f_n)}{(g_1 g_2 ... g_m)}, then:

\frac{q^\prime}{q} = (\frac{f_1^\prime}{f_1} + \frac{f_2^\prime}{f_2} + ... + \frac{f_n^\prime}{f_n}) - (\frac{g_1^\prime}{g_1} + \frac{g_2^\prime}{g_2} + ... + \frac{g_m^\prime}{g_m})

I should mention that this generalized formula is a simple application of logarithmic differentiation.  This gives credence to the notion that the truly important differentiation rule is the chain rule.  Everything else easily follows from that.

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