Page:Calculus Made Easy.pdf/37



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:

$$\frac{d^2y}{dx^2}$$; 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 $$x$$.