Notes and Reflections on Books and Media
by Hannah Leitheiser
Calculus Made Easy
Silvanus P. THOMPSON
Audiobooks in mathematics are a bit of a challenge, because my training -- if you can call it that -- in mathematics has been visual.
This book takes a non-standard approach of considering orders of smallness in infinitesimals. As as simple example, let's take y = x^3. Now we can create an equation for the addition of a little bit to each variable:
(y + dy) = (x + dx)^3
y + dy = (x + dx)(x^2 + 2xdx + dx^2)
y + dy = x^3 + 2(x^2)dx + x(dx^2) + x^2dx + 2x(dx^2) + dx^3
Now we make the assertion that powers of dx or dy higher than one are generally orders of smallness that can be neglected when our concern is to find dx and dy. (A small thing times itself is a very small thing.) And so.
y + dy = x^3 + 2(x^2)dx + (x^2)dx
y - x^3 + dy = dx (3x^2)
x^3 - x^3 + dy = dx (3x^2)
dy = dx (3x^2)
And we arrive at the conclusion you'd expect:
dy/dx = 3x^2
At least in polynomials, it's basically the same as limits, since you'll find the limit as h approaches zero of things like xh^2/h is zero, and so it falls out.