Derivative of the Quotient: Lost in Translation?

I posted a while ago about an alternate way to calculate the derivative of a quotient. Suppose G and H are functions of one or more variables.  For F = G/H, the standard form of the first derivative is:

F’ = (HG’ – H’G)/H^2,

and the alternate form is

F’ = (G’ – F H’)/H.

There was one comment on the post:

Интересная статейка, но как по мне, можно было бы и глубже капнуть..)

I was pleased to see the first non-English comment on the blog. Since I don’t know Russian, I tried various free online translators, and this is what they came up with:

Interesting статейка but as on me, it would be possible and to drip more deeply

an interesting six million was totally unfounded, but as for me, could be and deeper be loud

Interesting [stateyka], but as on me, it would be possible and more deeply to drop

An interesting little article, but as for me, could be deeper and drip

It looks like the machine translators aren’t quite ready to put the humans out of business.

I asked my wife about the translation since she has studied Russian. She says that the point is to dig deeper into the topic.

So let’s dig deeper:

Again, the standard form of the derivative of F = G/H is (HG’ – H’G)/H^2 and the alternate form is (G’ – F H’)/H. We can compare the two forms as follows:

1. Which form is more helpful in finding a suitable symbolic expression for the derivative? The standard form is fine in many cases, but if you are getting bogged down with a very complicated problem (and assuming you aren’t using a program such as Mathematica to find it for you), you might try the alternate form.

2. Which is a faster way to compute the values of the derivative, given the values for F, G’, and H’? Perhaps you aren’t as interested in finding a nice expression for F’ as in just being able to compute its values. If you already have a way to compute F, G’, and H’, then the alternate form is faster because it uses two fewer operations.

3. Which is easier to remember? The standard formula is so symmetric that I find it hard to remember and I always have to look it up.  I can remember the alternate form.

4. Which better explains the relationship between F and F’? The alternate form is more helpful since it explicitly includes F. You could work out the when F and F’ have the same sign. It is easy to see that the critical points (the maxima, minima, and inflection points) of F occur wherever F = G’/H’, since this condition is equivalent to G’ = F H’ and F’ = 0.  This property applies only to functions of a single real variable.

I hadn’t known the elegant property that the critical points of a quotient function are found where F = G’/H’.  I tried it out for various quotient functions, including:

f(x) = sin(x)/x for x <> 0, and f(x) = 1 for x = 0.

Since the derivative of sin(x) is cos(x) and the derivative of x is 1, the critical points of f should occur wherever f(x) = cos(x). The graph below shows that the local maxima and minima of f are found where f(x) and cos(x) intersect:

More generally, if f(x) = g(x)/x for x <> 0, then the critical points of f will be found wherever f(x) = g’(x).

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