Page:Calculus Made Easy.pdf/37

NOTE TO CHAPTER III.

How to read Differentials.

It will never do to fall into the schoolboy error of thinking that $dx$ means $d$ times $x$, for $d$ is not a factor–it means “an element of” or “a bit of” whatever follows. One reads $dx$ thus: “dee-eks.”

In case the reader has no one to guide him in such matters it may here be simply said that one reads differential coefficients in the following way. The differential coefficient

${\frac {dy}{dx}}$ is read “dee-wy by dee-eks,” or “dee-wy over dee-eks.”

So also ${\frac {du}{dt}}$ is read “dee-you by dee-tee.”

Second differential coefficients will be met with later on. They are like this:

${\frac {d^{2}y}{dx^{2}}}$; which is read “dee-two-wy over dee-eks-squared,” and it means that the operation of differentiating $y$ with respect to $x$ has been (or has to be) performed twice over.

Another way of indicating that a function has been differentiated is by putting an accent to the symbol of the function. Thus if $y=F(x)$, which means that $y$ is some unspecified function of $x$ (see p. 14), we may write $F'(x)$ instead of ${\frac {d(F(x))}{dx}}$. Similarly, $F''(x)$ will mean that the original function $F(x)$ has been differentiated twice over with respect to $x$.