Adrian McMenamin

A better demonstration of the product rule

Inspired by The Theoretical Minimum: What You Need to Know to Start Doing Physics: here’s a better proof/justification for the product rule in differential calculus than the one I set out here last month.

We will start with what we will treat as an axiomatic definition of the differential of the function y=f(x):

\frac{dy}{dx} = \frac{df(x)}{dx} = \frac{f(x+\Delta x) - f(x)}{\Delta x} as \Delta x \rightarrow 0

In this case we have y=f(x)g(x), so \frac{dy}{dx} = \frac{f(x + \Delta x)g(x +\Delta x) - f(x)g(x)}{\Delta x}

From our definition we can substitute for f(x+\Delta x) and g(x + \Delta x) and simplifying our notation for presentational reasons so that \frac{df(x)}{dx} = f^{\prime} etc:

f(x+\Delta x) = f^{\prime}\Delta x + f(x)

g(x+\Delta x) = g^{\prime}\Delta x + g(x)

Giving (after dividing through by \Delta x ):

y^{\prime} =f^{\prime}g^{\prime}\Delta x + g(x)f^{\prime} + \frac{f(x)g(x)}{\Delta x} + g^{\prime}f(x) - \frac{f(x)g(x)}{\Delta x}

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

As \Delta x \rightarrow 0 the first term falls to zero and so we are left with:

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

Which, of course, is the product rule.

Update: See this most excellent comment from Professor Rubin.

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2 comments

  1. Politicians (“tax revenues [cough][cough] equal [cough][cough] public expenditures”) and IT managers (“project launch date [cough][cough] equals [cough][cough] target date”) share a definition of “equals” different from the mathematical one. Unfortunately, that also applies to the line f(x + \Delta x) = f'(x)\Delta x + f(x), where “=” only means “approximates for small \Delta x“. Using “little-oh” notation (https://en.wikipedia.org/wiki/Big_O_notation#Little-o_notation), you can clean this up a bit. f(x + \Delta x) = f'(x)\Delta x + f(x) + o(\Delta x), and similarly for g(). All the o(\Delta x) terms (and products of them) can be gathered into a single composite o(\Delta x) term which, after division by \Delta x, dies at the limit.

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