We have already explained that what we call a derivative is often called a differential coefficient. Not only a different name but a different notation is often used; the derivative of the function $$y = \phi(x)$$ is often denoted by one or other of the expressions $D_{x}y,\quad \frac{dy}{dx}.$ Of these the last is the most usual and convenient: the reader must however be careful to remember that $$dy/dx$$ does not mean ‘a certain number $$dy$$ divided by another number $$dx$$’: it means ‘the result of a certain operation $$D_{x}$$ or $$d/dx$$ applied to $$y = \phi(x)$$’, the operation being that of forming the quotient $$\{\phi(x + h) – \phi(x)\}/h$$ and making $$h \to 0$$.

Of course a notation at first sight so peculiar would not have been adopted without some reason, and the reason was as follows. The denominator $$h$$ of the fraction $$\{\phi(x + h) – \phi(x)\}/h$$ is the difference of the values $$x+h$$, $$x$$ of the independent variable $$x$$; similarly the numerator is the difference of the corresponding values $$\phi(x + h)$$, $$\phi(x)$$ of the dependent variable $$y$$. These differences may be called the increments of $$x$$ and $$y$$ respectively, and denoted by $$\delta x$$ and $$\delta y$$. Then the fraction is $$\delta y/\delta x$$, and it is for many purposes convenient to denote the limit of the fraction, which is the same thing as $$\phi'(x)$$, by $$dy/dx$$. But this notation must for the present be regarded as purely symbolical. The $$dy$$ and $$dx$$ which occur in it cannot be separated, and standing by themselves they would mean nothing: in particular $$dy$$ and $$dx$$ do not mean $$\lim\delta y$$ and $$\lim\delta x$$, these limits being simply equal to zero. The reader will have to become familiar with this notation, but so long as it puzzles him he will be wise to avoid it by writing the differential coefficient in the form $$D_{x}y$$, or using the notation $$\phi(x)$$, $$\phi'(x)$$, as we have done in the preceding sections of this chapter.

In Ch. VII, however, we shall show how it is possible to define the symbols $$dx$$ and $$dy$$ in such a way that they have an independent meaning and that the derivative $$dy/dx$$ is actually their quotient.

The theorems of § 113 may of course at once be translated into this notation. They may be stated as follows:

(1) if $$y = y_{1} + y_{2}$$, then $\frac{dy}{dx} = \frac{dy_{1}}{dx} + \frac{dy_{2}}{dx};$

(2) if $$y = ky_{1}$$, then $\frac{dy}{dx} = k\frac{dy_{1}}{dx};$

(3) if $$y = y_{1}y_{2}$$, then $\frac{dy}{dx} = y_{1}\frac{dy_{2}}{dx} + y_{2}\frac{dy_{1}}{dx};$

(4) if $$y = \dfrac{1}{y_{1}}$$, then $\frac{dy}{dx} = -\frac{1}{y_{1}^{2}}\, \frac{dy_{1}}{dx};$

(5) if $$y = \dfrac{y_{1}}{y_{2}}$$, then $\frac{dy}{dx} = \biggl(y_{2}\frac{dy_{1}}{dx} – y_{1}\frac{dy_{2}}{dx}\biggr) \bigg/ y_{2}^{2};$

(6) if $$y$$ is a function of $$x$$, and $$z$$ a function of $$y$$, then $\frac{dz}{dx} = \frac{dz}{dy}\, \frac{dy}{dx};$

Example XL

1. If $$y = y_{1}y_{2}y_{3}$$ then $\frac{dy}{dx} = y_{2}y_{3}\, \frac{dy_{1}}{dx} + y_{3}y_{1}\, \frac{dy_{2}}{dx} + y_{1}y_{2}\, \frac{dy_{3}}{dx},$ and if $$y = y_{1}y_{2} \dots y_{n}$$ then $\frac{dy}{dx} = \sum_{r=1}^{n} y_{1}y_{2} \dots y_{r-1}y_{r+1} \dots y_{n}\, \frac{dy_{r}}{dx}.$ In particular, if $$y = z^{n}$$, then $$dy/dx = nz^{n-1}(dz/dx)$$; and if $$y = x^{n}$$, then $$dy/dx = nx^{n-1}$$, as was proved otherwise in EX. XXXIX. 3.

2. If $$y = y_{1}y_{2}\dots y_{n}$$ then $\frac{1}{y}\, \frac{dy}{dx} = \frac{1}{y_{1}}\, \frac{dy_{1}}{dx} + \frac{1}{y_{2}}\, \frac{dy_{2}}{dx} + \dots + \frac{1}{y_{n}}\, \frac{dy_{n}}{dx}.$ In particular, if $$y = z^{n}$$, then $$\dfrac{1}{y}\, \dfrac{dy}{dx} = \dfrac{n}{z}\, \dfrac{dz}{dx}$$.