## The Product and Quotient Rules for Differential Calculus

A couple of weeks ago one of my senior colleagues subbed for me on the day we discussed the product and quotient rules for differential calculus.  Afterwards he told me that he had never seen the way that I introduced these formulas, so I thought I would reproduce the derivations here. They are not original, but I cannot remember now where I first saw them.  The product rule derivation is more standard, I think.  The quotient rule derivation is nice because it avoids the chain rule.

The Product Rule.

The product rule tells us how to evaluate $\dfrac{d}{dx} f(x)g(x)$.

To understand this it helps to visualize what is going on.  You can think of the product as representing the area of a rectangle with sides $f(x)$ and $g(x)$.  If we increase $x$ by $\Delta x$ we are (if f and g are increasing at $x$) making tiny increases in the side lengths of the rectangle.  Effectively, we are adding the three shaded areas (two blue and one gray) to the white rectangle in the picture below.  (Image borrowed from the Wikipedia article on the product rule.)

The blue shaded area on the top is $\Delta f(x) g(x)$, the blue shaded area on the right is $f(x) \Delta g(x)$, and the gray shaded area in the corner is $\Delta f(x) \Delta g(x)$.

With this in mind, let’s take the limit of the difference quotient in order to calculate the derivative. This gives

$\displaystyle \dfrac{d}{dx} f(x)g(x) = \lim_{\Delta x \to 0} \frac{\Delta f(x) g(x) + f(x) \Delta g(x) + \Delta f(x) \Delta g(x)}{\Delta x}$
$\displaystyle = g(x) \lim_{\Delta x \to 0} \frac{\Delta f(x)}{\Delta x} + f(x) \lim_{\Delta x \to 0} \frac{\Delta g(x)}{\Delta x} + \lim_{\Delta x \to 0} \frac{\Delta f(x) \Delta g(x)}{\Delta x}$
$\displaystyle = g(x) f'(x) + f(x) g'(x) + f'(x)(0)$
$= g(x) f'(x) + f(x) g'(x)$, which is the product rule.

(This argument is essentially the one on the Wikipedia page for the product rule.)

The Quotient Rule.

The quotient rule tells us how to evaluate $\dfrac{d}{dx} \dfrac{f(x)}{g(x)}$.

A standard way to derive the quotient rule is to use the product rule together with the chain rule.  However, by the point in the semester at which I want to introduce the quotient rule we generally haven’t seen the chain rule yet.  Here is a derivation that avoids the chain rule.

Let $\displaystyle Q(x) = \frac{f(x)}{g(x)}$.  We want $Q'(x)$.  Algebra gives us $f(x) = Q(x)g(x)$.  Differentiating both sides, we have $f'(x) = g(x) Q'(x) + Q(x)g'(x)$, by the product rule.  Now we solve for $Q'(x)$ to obtain the quotient rule:

$\displaystyle Q'(x) = \dfrac{f'(x) - Q(x) g'(x)}{g(x)} = \dfrac{f'(x) - \frac{f(x)}{g(x)}g'(x)}{g(x)} = \dfrac{\frac{g(x)}{g(x)}f'(x) - \frac{f(x)}{g(x)} g'(x)}{g(x)} = \dfrac{g(x) f'(x) - f(x)g'(x)}{(g(x))^2}.$