I’ve gotten back into my quest to learn calculus, but I’ve run into a little hiccup in the form of what appears to be a fundamental error in the book I’m using (Silvanus P. Thompson’s *Calculus Made Easy*, Second Edition (the PDF version at that link).

On page 18, we see this paragraph:

What does (dx)^2 mean? Remember that dx meant a bit – a little bit – of x. Then (dx)^2 will mean a little bit of a little bit of x; that is, as explained above (p. 4), it is a small quantity of the second order of smallness.

It may therefore be discarded as quite inconsiderable in comparison with the other terms.

Emphasis mine. This is the part that really made me go “Huh?”, but I figured that the author knew more than I do, and decided to leave it.

Then later, on pages 19 & 20, we are given an example of differentiating y=x^3, and just to make sure I was understanding things correctly I tried to work it out. I decided I would have x=3 and dx=1. Here is the example from the text:

This ends with *dy/dx = 3x^2*, but that part won’t copy for some reason.

But, there’s a problem. If, as I did, you set *x=3* and *dx*=1*, *then you end up with *y=27* and *y+dy=64*, making *dy=37*. But, if you take *(3x^2)(dx)* when *x=3* and *dx=1*, you end up with *dy=27*.

So, I tried working it out without discarding *(dx)^2* and *(dx)^3*, and got:

*y + dy = x ^{3} + 3x^{2}(dx) + 3x(dx)^{2} + (dx)^{3}*

*y = x ^{3}*, so they cancel and give:

*dy = 3x ^{2}(dx) + 3x(dx)^{2} + (dx)^{3}*

*dy = 3(3 ^{2})(1) + 3(3)(1^{2}) + (1)^{3}*

*dy = 27 + 9 + 1 = 37*

The right answer.

A 10 out of 37 error is *not* negligible as the book asserts. So, can anyone who knows calculus tell me who is wrong here? Me, or the book? And why?

Help!

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