# 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.

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.