110-112. Derivatives

110. Derivatives or Differential Coefficients.

Let us return to the consideration of the properties which we naturally associate with the notion of a curve. The first and most obvious property is, as we saw in the last chapter, that which gives a curve its appearance of connectedness, and which we embodied in our definition of a continuous function.

The ordinary curves which occur in elementary geometry, such as straight lines, circles and conic sections, have of course many other properties of a general character. The simplest and most noteworthy of these is perhaps that they have a definite direction at every point, or what is the same thing, that at every point of the curve we can draw a tangent to it. The reader will probably remember that in elementary geometry the tangent to a curve at $P$ is defined to be ‘the limiting position of the chord $PQ$, when $Q$ moves up towards coincidence with $P$’. Let us consider what is implied in the assumption of the existence of such a limiting position.

In the figure (Fig. 36) $P$ is a fixed point on the curve, and $Q$ a variable point; $PM$, $QN$ are parallel to $OY$ and $PR$ to $OX$. We denote the coordinates of $P$ by $x$, $y$ and those of $Q$ by $x + h$, $y + k$: $h$ will be positive or negative according as $N$ lies to the right or left of $M$.

We have assumed that there is a tangent to the curve at $P$, or that there is a definite ‘limiting position’ of the chord $PQ$. Suppose that $PT$, the tangent at $P$, makes an angle $\psi$ with $OX$. Then to say that $PT$ is the limiting position of $PQ$ is equivalent to saying that the limit of the angle $QPR$ is $\psi$, when $Q$ approaches $P$ along the curve from either side. We have now to distinguish two cases, a general case and an exceptional one.

The general case is that in which $\psi$ is not equal to $\frac{1}{2}\pi$, so that $PT$ is not parallel to $OY$. In this case $RPQ$ tends to the limit $\psi$, and \[RQ/PR = \tan RPQ\] tends to the limit $\tan \psi$. Now \[RQ/PR = (NQ – MP)/MN = \{\phi(x + h) – \phi(x)\}/h;\] and so \[\begin{equation*}\lim_{h \to 0} \frac{\phi(x + h) – \phi(x)}{h} = \tan\psi. \tag{1}\end{equation*}\]

The reader should be careful to note that in all these equations all lengths are regarded as affected with the proper sign, so that (e.g.) $RQ$ is negative in the figure when $Q$ lies to the left of $P$; and that the convergence to the limit is unaffected by the sign of $h$.

Thus the assumption that the curve which is the graph of $\phi(x)$ has a tangent at $P$, which is not perpendicular to the axis of $x$, implies that $\phi(x)$ has, for the particular value of $x$ corresponding to $P$, the property that $\{\phi(x + h) – \phi(x)\}/h$ tends to a limit when $h$ tends to zero.

This of course implies that both of \[\{\phi(x + h) – \phi(x)\}/h,\quad \{\phi(x – h) – \phi(x)\}/(-h)\] tend to limits when $h \to 0$ by positive values only, and that the two limits are equal. If these limits exist but are not equal, then the curve $y = \phi(x)$ has an angle at the particular point considered, as in Fig. 37.

Now let us suppose that the curve has (like the circle or ellipse) a tangent at every point of its length, or at any rate every portion of its length which corresponds to a certain range of variation of $x$. Further let us suppose this tangent never perpendicular to the axis of $x$: in the case of a circle this would of course restrict us to considering an arc less than a semicircle. Then an equation such as (1) holds for all values of $x$ which fall inside this range. To each such value of $x$ corresponds a value of $\tan\psi$: $\tan\psi$ is a function of $x$, which is defined for all values of $x$ in the range of values under consideration, and which may be calculated or derived from the original function $\phi(x)$. We shall call this function the or derived function of $\phi(x)$, and we shall denote it by \[\phi'(x).\]

Another name for the derived function of $\phi(x)$ is the differential coefficient of $\phi(x)$; and the operation of calculating $\phi'(x)$ from $\phi(x)$ is generally known as differentiation. This terminology is firmly established for historical reasons: see § 115.

Before we proceed to consider the special case mentioned above, in which $\psi = \frac{1}{2}\pi$, we shall illustrate our definition by some general remarks and particular illustrations.

111. Some general remarks.

(1) The existence of a derived function $\phi'(x)$ for all values of $x$ in the interval $a \leq x \leq b$ implies that $\phi(x)$ is continuous at every point of this interval. For it is evident that $\{\phi(x + h) – \phi(x)\}/h$ cannot tend to a limit unless $\lim\phi(x + h) = \phi(x)$, and it is this which is the property denoted by continuity.

(2) It is natural to ask whether the converse is true, i.e. whether every continuous curve has a definite tangent at every point, and every function a differential coefficient for every value of $x$ for which it is continuous.¹ The answer is obviously No: it is sufficient to consider the curve formed by two straight lines meeting to form an angle (Fig. 37). The reader will see at once that in this case $\{\phi(x + h) – \phi(x)\}/h$ has the limit $\tan\beta$ when $h \to 0$ by positive values and the limit $\tan\alpha$ when $h \to 0$ by negative values.

This is of course a case in which a curve might reasonably be said to have two directions at a point. But the following example, although a little more difficult, shows conclusively that there are cases in which a continuous curve cannot be said to have either one direction or several directions at one of its points. Draw the graph (Fig. 14, § 28) of the function $x\sin(1/x)$. The function is not defined for $x = 0$, and so is discontinuous for $x = 0$. On the other hand the function defined by the equations \[\phi(x) = x\sin(1/x)\quad (x \neq 0),\qquad \phi(x) = 0\quad (x = 0)\] is continuous for $x = 0$ (Exs. XXXVII. 14, 15), and the graph of this function is a continuous curve.

But $\phi(x)$ has no derivative for $x = 0$. For $\phi'(0)$ would be, by definition, $\lim\{\phi(h) – \phi(0)\}/h$ or $\lim\sin(1/h)$; and no such limit exists.

It has even been shown that a function of $x$ may be continuous and yet have no derivative for any value of $x$, but the proof of this is much more difficult. The reader who is interested in the question may be referred to Bromwich’s Infinite Series, pp. 490–1, or Hobson’s Theory of Functions of a Real Variable, pp. 620–5.

(3) The notion of a derivative or differential coefficient was suggested to us by geometrical considerations. But there is nothing geometrical in the notion itself. The derivative $\phi'(x)$ of a function $\phi(x)$ may be defined, without any reference to any kind of geometrical representation of $\phi(x)$, by the equation \[\phi'(x) = \lim_{h \to 0} \frac{\phi(x + h) – \phi(x)}{h};\] and $\phi(x)$ has or has not a derivative, for any particular value of $x$, according as this limit does or does not exist. The geometry of curves is merely one of many departments of mathematics in which the idea of a derivative finds an application.

Another important application is in dynamics. Suppose that a particle is moving in a straight line in such a way that at time $t$ its distance from a fixed point on the line is $s = \phi(t)$. Then the ‘velocity of the particle at time $t$’ is by definition the limit of \[\frac{\phi(t + h) – \phi(t)}{h}\] as $h \to 0$. The notion of ‘velocity’ is in fact merely a special case of that of the derivative of a function.

Example XXXIX

1. If $\phi(x)$ is a constant then $\phi'(x) = 0$. Interpret this result geometrically.

2. If $\phi(x) = ax + b$ then $\phi'(x) = a$. Prove this (i) from the formal definition and (ii) by geometrical considerations.

3. If $\phi(x) = x^{m}$, where $m$ is a positive integer, then $\phi'(x) = mx^{m-1}$.

[For \[\begin{aligned} \phi'(x) &= \lim \frac{(x + h)^{m} – x^{m}}{h}\\ &= \lim \left\{mx^{m-1} + \frac{m(m – 1)}{1\cdot2} x^{m-2} h + \dots + h^{m-1}\right\}.\end{aligned}\]

The reader should observe that this method cannot be applied to $x^{p/q}$, where $p/q$ is a rational fraction, as we have no means of expressing $(x + h)^{p/q}$ as a finite series of powers of $h$. We shall show later on (§ 118) that the result of this example holds for all rational values of $m$. Meanwhile the reader will find it instructive to determine $\phi'(x)$ when $m$ has some special fractional value (eg.. $\frac{1}{2}$), by means of some special device.]

4. If $\phi(x) = \sin x$, then $\phi'(x) = \cos x$; and if $\phi(x) = \cos x$, then $\phi'(x) = -\sin x$.

[For example, if $\phi(x) = \sin x$, we have \[\{\phi(x + h) – \phi(x)\}/h = \{2\sin \tfrac{1}{2}h \cos(x + \tfrac{1}{2}h)\}/h,\] the limit of which, when $h \to 0$, is $\cos x$, since $\lim\cos(x + \frac{1}{2}h) = \cos x$ (the cosine being a continuous function) and $\lim\{(\sin \frac{1}{2}h)/\frac{1}{2}h\} = 1$ (Ex. xxxvi. 13).]

5. Equations of the tangent and normal to a curve $y = \phi(x)$. The tangent to the curve at the point $(x_{0}, y_{0})$ is the line through $(x_{0}, y_{0})$ which makes with $OX$ an angle $\psi$, where $\tan\psi = \phi'(x_{0})$. Its equation is therefore \[y – y_{0} = (x – x_{0}) \phi'(x_{0});\] and the equation of the normal (the perpendicular to the tangent at the point of contact) is \[(y – y_{0}) \phi'(x_{0}) + x – x_{0} = 0.\] We have assumed that the tangent is not parallel to the axis of $y$. In this special case it is obvious that the tangent and normal are $x = x_{0}$ and $y = y_{0}$ respectively.

6. Write down the equations of the tangent and normal at any point of the parabola $x^{2} = 4ay$. Show that if $x_{0} = 2a/m$, $y_{0} = a/m^{2}$, then the tangent at $(x_{0}, y_{0})$ is $x = my + (a/m)$.

112.

We have seen that if $\phi(x)$ is not continuous for a value of $x$ then it cannot possibly have a derivative for that value of $x$. Thus such functions as $1/x$ or $\sin(1/x)$, which are not defined for $x = 0$, and so necessarily discontinuous for $x = 0$, cannot have derivatives for $x = 0$. Or again the function $[x]$, which is discontinuous for every integral value of $x$, has no derivative for any such value of $x$.

Example. Since $[x]$ is constant between every two integral values of $x$, its derivative, whenever it exists, has the value zero. Thus the derivative of $[x]$, which we may represent by $[x]’$, is a function equal to zero for all values of $x$ save integral values and undefined for integral values. It is interesting to note that the function $1 – \dfrac{\sin\pi x}{\sin\pi x}$ has exactly the same properties.

We saw also in Exs. XXXVII. 7 that the types of discontinuity which occur most commonly, when we are dealing with the very simplest and most obvious kinds of functions, such as polynomials or rational or trigonometrical functions, are associated with a relation of the type \[\phi(x) \to +\infty\] or $\phi(x) \to -\infty$. In all these cases, as in such cases as those considered above, there is no derivative for certain special values of $x$. In fact, as was pointed out in § 111, (1), all discontinuities of $\phi(x)$ are also discontinuities of $\phi'(x)$. But the converse is not true, as we may easily see if we return to the geometrical point of view of § 110 and consider the special case, hitherto left aside, in which the graph of $\phi(x)$ has a tangent parallel to $OY$. This case may be subdivided into a number of cases, of which the most typical are shown in Fig. 38. In cases (c) and (d) the function is two valued on one side of $P$ and not defined on the other. In such cases we may consider the two sets of values of $\phi(x)$, which occur on one side of $P$ or the other, as defining distinct functions $\phi_{1}(x)$ and $\phi_{2}(x)$, the upper part of the curve corresponding to $\phi_{1}(x)$.

The reader will easily convince himself that in (a) \[\{\phi(x + h) – \phi(x)\}/h \to +\infty,\] as $h \to 0$, and in (b) \[\{\phi(x + h) – \phi(x)\}/h \to -\infty;\] while in (c) \[\{\phi_{1}(x + h) – \phi_{1}(x)\}/h \to +\infty,\quad \{\phi_{2}(x + h) – \phi_{2}(x)\}/h \to -\infty,\] and in (d) \[\{\phi_{1}(x + h) – \phi_{1}(x)\}/h \to -\infty,\quad \{\phi_{2}(x + h) – \phi_{2}(x)\}/h \to +\infty,\] though of course in (c) only positive and in (d) only negative values of $h$ can be considered, a fact which by itself would preclude the existence of a derivative.

We can obtain examples of these four cases by considering the functions defined by the equations \[(a)\ y^{3} = x,\quad (b)\ y^{3} = -x,\quad (c)\ y^{2} = x,\quad (d)\ y^{2} = -x,\] the special value of $x$ under consideration being $x = 0$.

We leave out of account the exceptional case (which we have still to examine) in which the curve is supposed to have a tangent perpendicular to $OX$: apart from this possibility the two forms of the question stated above are equivalent.↩︎

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113. General rules for differentiation $\rightarrow$