Calculus Made Easy

“Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.”

“Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics—and they are mostly clever fools—seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.”

“Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.”

“The preliminary terror, which chokes off most [students]... from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning—in common sense terms—of the two principal symbols that are used in calculating.”

“(1) d... merely means "a little bit of." Thus dx means a little bit of x or du means a little bit of u. ...you will find that these little bits or elements may be considered to be indefinitely small.”

“(2) means the sum of all the little bits of t. Ordinary mathematicians call this symbol "the integral of."”

“Now any fool can see that if x is considered as made up of a lot of little bits, each of which is called dx, if you add them all up together you get the sum of all the dxs (which is the same thing as the whole of x). The word "integral" simply means "the whole."”

“When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow. That's all.”

“We shall have... to learn under what circumstances we may consider small quantities to be so minute that we may omit them from consideration. Everything depends upon relative minuteness.”

“If, for the purpose of time, 1/60 be called a small fraction, then 1/60 of 1/60 (being a small fraction of a small fraction) may be regarded as a small quantity of the second order of smallness.”

“The smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become. Hence we know that in all cases we are justified in neglecting the small quantities of the second—or third (or higher)—orders, if only we take the small quantity of the first order small enough in itself.”

“It must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred.”

“Let us think of x as a quantity that can grow by a small amount so as to become .”

“We call the ratio dy/dx "the differential coefficient of y with respect to x." It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy. Instead of being frightened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names; and, having relieved our minds, will go on to the simple thing itself, namely the ratio dy/dx.”

“The process of finding the value of dy/dx is called "differentiating." But, remember, what is wanted is the value of this ratio when both dy and dx are themselves indefinitely small. The true value of the differential coefficient is that to which it approximates in the limiting case when each of them is considered as infinitesimally minute.”

“Let us collect the results to see if we can infer any general rule...”

“Following out logically our observation, we should conclude that if we want to deal with any higher power,—call it n—we could tackle it in the same way. Let y x.”