3
Differential calculus
The introduction of the infinitesimal calculus, independently by Newton and Leibnitz in the late seventeenth century, was one of the most important events not only in the history of mathematics but also of physics, where it has been an indispensable tool ever since.
In this chapter and the one that follows, we introduce the formalism in the context of functions of a single variable. We start by considering differentiation, the calculation of the instantaneous rate of change of a function as its argument changes. So, for example, given a function x(t), which specifies the position of a particle moving in one dimension as a function of time t, the operation of differentiation yields a function 
representing the velocity. The inverse operation, called integration, will be discussed in Chapter 4 and enables the position x(t) to be deduced from 
and the value of x at some time, for example t = 0. These two operations – differentiation and integration – play a crucial role in understanding not only mechanics, but the whole of physical science. Both rest on ideas of limits and continuity, to which we now turn.
3.1 Limits and continuity
In previous chapters, we have used the ideas of limits and continuity in simple cases where their meaning is obvious. In this section we shall define them more precisely, before showing how they lead naturally to the idea of differentiation in Section 3.2
3.1.1 Limits
If a function f(x) approaches arbitrarily close to a fixed value α as x approaches arbitrarily close to a constant a, then α is said to be the limit of f(x) as x approaches a, with the notation
(3.1a) 
or equivalently
(3.1b) 
More precisely, (3.1) means that for any ϵ > 0, however small, we can always find a number δ > 0, depending on ϵ, such that
For example, the obvious result
is formally verified by noting that f(x) = (x − 1)2 + 2, so that
and thus for any ϵ, however small, |f(x) − 2| < ϵ, provided that 
.
In this example, the limit of f(x) as x → 1 is equal to the value at x = 1, i.e.
However, the existence of the limit (3.1) does not in general imply that f(a) = α, and indeed f(a) may not even exist. For example, consider the function
Taking the limit as x → 3 gives
(3.3b) 
because
However, direct evaluation of (3.3a) at x = 3, gives f(3) = 0/0 and is undefined.
A number of important results follow directly from the definition of a limit. With the notations
and taking c as a constant, these are
- if f(x) = c, then
; 
;
;
;
;and, if n is an integer,
, if α1/n is defined.
The proof of these results is straightforward. As an example, we will prove (iv) as follows. From the definition (3.2) and (3.4),
Let δ1 < δ2. Then
If ϵ is chosen to be the positive root of
where η is any small quantity, then (3.5) may be written
which concludes the proof.
The definition of a limit can be extended to the case where x increases indefinitely, either positively or negatively. For example,
means that, for any ϵ > 0, however small, a number l > 0 can be found such that |f(x) − α| < ϵ for any x > l. If f(x) increases indefinitely, positively or negatively, as x → a, we will use the notation
with the appropriate sign.
The following examples illustrate these results.
3.1.2 Continuity
So far, we have not specified the path taken as x → a. There are two possibilities. Firstly, x could tend to a via values less than a. This is referred to as approaching a from the left (or below) and the limit is denoted 
Alternatively, if x approaches a via values greater than a, then x is said to approach a from the right (or above) and the limit is written 
For the limit (3.1) to exist, these two limits must be identical, since the defining condition (3.2) is independent of the sign of (x − a). However, in practice, the two limits are not always the same. As an example, consider the function 
. At x = 0, f(x) is undefined and in addition
(3.7a) 
but
(3.7b) 
In general, a function f(x) is said to be continuous at the point x = x0 if the following conditions are satisfied:
If a function f(x) is defined in the interval (a, x0) to the left of x0, and in the interval (x0, b) to the right of x0, then f(x) is said to be discontinuous at x0 if either of the above conditions fails at x = x0. Thus the function 
above is said to be discontinuous at the point x = 0, as shown in Figure 3.1. Another example is the function
In this case, 
and is undefined, and f(x) is discontinuous at x = 3. However, we saw in (3.4b) that in the limit as x → 3, f(x) → 6, so that in this case we could define a function g(x) that is identical to f(x) except at x = 3, where we define g(x) to be 6. Then the function g(x) would be continuous. This type of discontinuity, which can be removed by redefining the value of the function at the point of discontinuity, is said to be removable. In the case of the function 
at x = 0, the function is discontinuous because the limits from above and below are not equal (cf. Eqs. 3.7). This is called a jump discontinuity and is not removable. Another type of jump discontinuity is illustrated by the plot of tan θ, shown in Figure 2.8(a). At 
, for example, tan θ is ill defined, since
while tan θ → ∞ as 
and to − ∞ as 
. This type of discontinuity, associated with divergent behaviour, is called an infinite discontinuity.
Figure 3.1 The function f(x) = x/|x| in the vicinity of x = 0.
It follows from the properties of limits discussed in Section 3.1.1, that the sum, product, difference or quotient of two functions that are both continuous at a point are themselves continuous, provided in the case of a quotient that the denominator does not vanish at the point.
3.2 Differentiation
The aim in this section is, given a function f(x), to find a function that gives the gradient of f(x) at a given value of the independent variable x. This is achieved by a limiting procedure. Consider the change in the function in going from x to x + δx, where δx is a small quantity, positive or negative, in a region where the function is continuous. Then the average rate of change of f(x) in the range x to x + δx is clearly
(3.9) 
and the instantaneous rate of change at x, denoted by 
, is given by
provided the limit exists. In this case, the function is said to be differentiable and 
is called the derivative of f(x) with respect to x. In calculating it, we say that we have differentiated f(x) with respect to x. This is illustrated in Figure 3.2, from which we see that the term in square brackets in (3.10) is just the gradient of the straight line AB, which approaches the gradient of the tangent at x as δx → 0. In other words, the derivative at x is the gradient of the curve at x.
Figure 3.2 Geometrical interpretation of the derivative df/dx.
There are several equivalent notations used to denote a derivative. Each is convenient for different circumstances. If, as in Chapter 1, we introduce the dependent variable y = f(x), these are
(3.11a) 
It is also useful to define
(3.11b) 
for a change in f(x) corresponding to a change from x to (x + δx) in the independent variable, so that (3.10) becomes
3.2.1 Differentiability
Differentiability is closely related to continuity. A necessary condition for a function to be differentiable is that it must be continuous, since the limit (3.10) cannot exist unless
so that the continuity conditions (3.8) are automatically satisfied. However, this alone is not a sufficient condition and a continuous function is not necessarily differentiable. To see this, consider the function f(x) = |x|. This is continuous, even at x = 0, because
However, one easily verifies that the quantity in brackets in (3.10) is equal to +1 for all x > 0, but −1 for all x < 0. Hence f(x) is not differentiable at x = 0, since
But it is differentiable at x ≠ 0.
3.2.2 Some standard derivatives
Although (3.10) is the fundamental definition, it is not necessary to use it directly in most cases. Rather, one uses it to deduce the derivatives for a number of important standard functions. These are then used, together with general properties that follow from (3.10), to deduce the result for other cases, as we shall see.
Here we shall consider some of these standard derivatives starting with simple powers f(x) = xn, where n is any integer. From the binomial theorem (1.23) we have
where O[(δx)2] means terms that are at most of order (δx)2, that is, are proportional to (δx)2, and so can be neglected compared to terms that are linear in δx as δx → 0. Hence,
In other words
This result also holds when n is not an integer, as we shall show in Section 3.3.5.
Next we consider the more difficult case of f(x) = sin x. Using (2.36c), we have
and since cos [(2x + δx)/2] → cos x as δx → 0, we have
It remains to find the limit of the term in the brackets. This is done by the construction of Figure 3.3, from which we see that the ratio of the length of the line AB to the length of the arc AC is given by
where the angle is as usual measured in radians. We also see that, as θ → 0, the lengths of AB and AC tend to equality, giving the important result
(3.15) 
Applying this to (3.14), with 
, gives the desired result
A similar argument, left to the reader, leads to the result
Figure 3.3 Construction to find the limit of sin θ/θ as θ → 0.
Finally, in Section 2.3.2 we showed, using essentially the argument formulated more generally at the beginning of this section, that the slope or gradient of ex was ex (cf. Eq. 2.49). In the present notation, this result is written
3.3 General methods
Methods for differentiating other functions may be derived using general properties of derivatives that follow from their definition, together with standard results like (3.13)–(3.16).
Suppose we have a function of the general form
where a1, a2⋅⋅⋅aN are constants and f1(x), f2(x), …, fN(x) are differentiable functions. Then from (3.11) we easily see that
so that
by (3.12). Hence, for example, if
then
by (3.19), together with the standard derivatives (3.16) and (3.17) for sin x and cos x, respectively. Similarly, for an arbitrary polynomial of order N, i.e.
we have
In what follows, we shall introduce a series of general results analogous to (3.19) and illustrate their use in finding the derivatives of specific functions by examples.
3.3.1 Product rule
Consider a function of the form
where u(x) and u(x) are differentiable functions. Then
Since the last term is second order in small quantities,1 it can be neglected in taking the limit (3.11), which then gives the product rule
3.3.2 Quotient rule
We next consider a quotient
where u(x) and 
are again arbitrary differentiable functions and 
. Then
so that (3.12) gives the quotient rule
where in the denominator we have used the fact that 
as δx → 0 for any differentiable function 
. Setting u = 1 leads immediately to the reciprocal rule:
since clearly 
for any constant u.
3.3.3 Reciprocal relation
The derivatives of functions and their inverses are closely related. Consider a function y = f(x) and its inverse x = f− 1(y), where we assume that both f and f− 1 are differentiable functions in what follows.2 When x → x + δx, the function y → y + δy and
while similarly
Since δx → 0 corresponds to δy → 0 and vice versa, the trivial relation
leads immediately to the reciprocal relation
for differentiable functions. Since y = f(x) and x = f− 1(y), this can alternatively be written
(3.25) 
so that df/dx is easily obtained if the derivative of the inverse 
is known.
We will illustrate this for the important case of y = f(x) = ln x. Then,
so that
by (3.17) and
giving the important result
(3.26) 
3.3.4 Chain rule
We next consider a function y that is itself a function of a second function z(x), that is, y[z(x)], or more explicitly
where f and g are continuous, differentiable functions of x. For such functions, when x → x + δx there are corresponding changes z → z + δz, y → y + δy, such that δy, δz → 0 when δx → 0. Hence
i.e.
Equation (3.29) is called the chain rule. When used together with judiciously chosen substitutions, it is a key tool in evaluating derivatives, as we shall immediately illustrate.
3.3.5 More standard derivatives
In this section we obtain some more standard derivatives, this time involving logarithms and exponentials. We start by considering functions of the form y = ln f(x), which using (3.28) may be written
Hence the chain rule (3.29) gives
i.e.
Equation (3.30) is called a logarithmic derivative. If, for example, we choose f(x) = 3x2, (3.30) gives
Another class of functions is exp [f(x)], when (3.29) gives
For the simple case f(x) = −x, this gives
which, together with the corresponding result (3.18) for ex, enables the hyperbolic functions to be differentiated. In this way, starting from the definitions (2.57), and using (3.19), one obtains the standard results:
(3.32a) 
and
(3.32b) 
The corresponding result for tanh x,
(3.32c) 
follows from 
using the quotient rule (3.22).
Another important result that follows from (3.31) is
for any real number α. Previously we obtained this result for integer α = n. To establish it in general, we note that
which is of the form ef(x) with f(x) = αln x. Relation (3.31) then gives
The result (3.33) is the last of a set of ‘standard derivatives’ that we have derived in this and previous sections and which are extremely useful in calculating the derivatives of other functions, using the product, quotient and chain rules, and the reciprocal relation (3.24). They are listed in Table 3.1 for later convenience.
Table 3.1 Some standard derivatives
 |
3.3.6 Implicit functions
So far we have discussed the techniques available to differentiate explicit functions. Here we briefly extend the discussion to include functions defined implicitly as the solution of an equation, or by parametric forms.
In the latter cases, both x and y are defined in terms of a third variable, a parameter t, say. That is, by equations of the form
where we assume f(t) and g(t) are themselves continuous differentiable functions. For example, x and y could specify the positions of a point in a plane as a function of the time t. Equations (3.34) imply a functional relationship between x and y that can be written as the explicit function y = g[f− 1(x)] if the function f has an inverse. However, to find the derivative of y with respect to x, it is easier to note that if a small change δt leads to changes δx, δy in x and y, then the trivial relation
implies
(3.35) 
since δt → 0 implies δx, δy → 0 for continuous functions f, g.
Alternatively, a function might be defined implicitly as a solution of an equation of the form
where c is a constant. The derivative of y with respect to x can then be deduced from the equation
(3.37) 
which follows directly from (3.36).
3.4 Higher derivatives and stationary points
We have seen above how to differentiate a function y = f(x) to yield its derivative
This derivative itself is often a differentiable function, in which case it may also be differentiated to give a second derivative,
(3.38a) 
which, like the first derivative (cf. Eq. 3.9) can be written in the alternative forms
(3.38a) 
The first derivative 
specifies the gradient or instantaneous rate of change of the function y(x) at any given x. Similarly, the second derivative (3.38) gives the instantaneous rate of change of the gradient itself. So, for example, if
implying that the slope of x2 itself increases as x increases at a constant rate 2, independent of x.
If the second derivative is differentiable, one can similarly define a third derivative
(3.39) 
or, more generally, an nth derivative
(3.40a) 
provided that all the lower derivatives exist and are differentiable. Using ‘primes’ as superscripts, as in (3.38b), is impractical for the general case, and an alternative notation is
(3.40b) 
Such higher derivatives, with n ≥ 3, can be important in applications, as we shall see in Chapter 5. Here we shall give one worked example, which we will require later, and then describe an important application that depends on the first and second derivatives only.
3.4.1 Stationary points
In examining the form of a given function y = f(x), it is often useful to consider not only its roots defined by the requirement y = 0, but also the points x0 defined by the condition
These are called stationary points, because the instantaneous rate of change of f(x) with respect to x vanishes at x = x0, and the tangent to the curve is horizontal, as shown in Figure 3.4. The figure shows four types of stationary point, corresponding to different behaviours of the gradient f′(x) immediately below and immediately above the stationary point x = x0.
Local minima3
In this case, the gradient f′(x) is negative immediately below and positive immediately above the stationary point x = x0, as shown in Figure 3.4(a). Because f″(x) is the instantaneous rate of change of f′(x), and f′(x0) = 0, this implies
since otherwise f′(x) would be negative immediately above x = x0, in contradiction to our assumption. In other words, the existence of a local minimum at x = x0 implies
(3.43a)
Local maxima
In this case, the gradient of the function is positive immediately above and negative immediately below the stationary point, as shown in Figure 3.4(b). An argument similar to that given above for local maxima leads to
(3.43b)
Stationary points of inflection4
These correspond to the case where f′(x) has the same sign on both sides of the stationary point, and can be positive, as shown in Figure 3.4(c), or negative, as shown in Figure 3.4(d). Consider the first case, in which f′(x) is positive both immediately below and above x = x0. Since f′(x) = 0 at x = x0, it follows that x0 is a stationary point (a minimum) of f′(x), implying that its derivative f″(x0) = 0. A similar argument applies to Figure 3.4(d), corresponding to the case where f′(x) is negative on both sides of the stationary point, leading again to the result f″(x0) = 0. Hence for a stationary point of inflection5
(3.43c)
Figure 3.4 The behaviour of a function (solid line) and its derivative (dashed line) in the vicinity of a stationary point x = x0, together with the gradient at x0 (dotted line), for (a) a minimum, (b) a maximum and (c, d) points of inflection.
The three cases (i), (ii) and (iii), exhaust all possibilities for the signs of f′(x) in the immediate vicinity of the stationary point. To summarise, from (3.43) the conditions
at x = x0 unambiguously identifies the stationary point as a minimum and
(3.44b) 
unambiguously identifies the stationary point as a maximum. On the other hand, the combination
can correspond to a maximum, a minimum or a point of inflection and we must examine the behaviour of the derivative f′(x) on both sides of the stationary point to distinguish them.6
To illustrate this, consider the simple cases f(x) = x2, x3 and x4, which all have a stationary point at x = 0. In the first case, we have
which satisfies (3.44a) at x = 0, and so is a minimum. In the second case,
so that (3.44c) is satisfied at x = 0. However, f′(x) > 0 both immediately below and above x = 0, so that we have a point of inflection. In the final case we have
so that (3.44c) is again satisfied at x = 0. However, in this case f′(x) < 0 for x < 0 and f′(x) > 0 for x > 0, so that x = 0 is a minimum. These three functions are plotted in Figure 3.5, where their behaviours at the stationary point are clearly seen.
Figure 3.5 The functions f(x) = x2, x3 and x4.
3.5 Curve sketching
Curve sketching is a very useful way of understanding and summarising the main features of a given function y = f(x). When doing so, it is important to pay attention to
- the limiting behaviour of the function as x → ±∞,
- any roots, where y = 0,
- any stationary points, where f′(x) = 0,
as well as any other general features, for example if the function is symmetric or antisymmetric, or if there are any discontinuities.
In the rest of this section, we shall illustrate the above points by a series of examples. In so doing, we shall assume that the main features of the plots of sin x, cos x, ex and ln x, given in Figures 2.6 and 2.8, may be used without citation. These functions permeate the whole of physical science and their characteristic forms are well worth memorising.
Problems 3
-
Find the limits of
as (a) x → 0, (b) x → 1 (c) x → ∞.
-
Find the following limits:
-
If f(x) = x2, prove from first principles that
. (Hint: it is sufficient to prove Eq. 3.1 assuming that δ = |x − 2| < 1, and you may use the general relation |a + b| ≤ |a| + |b| for any a, b.) -
Find the locations x0 of any discontinuities in the following functions and classify them as removable or non-removable. In the former case, specify the redefined value f(x0) required to remove the disconti- nuity.
-
Identify the locations x0 of any discontinuities in the following functions and classify them as removable or non-removable. In the case of removable discontinuities, find the redefined value f(x0) required to remove the discontinuity.
-
Consider the function
where A, B are constants and the integer n ≥ 1. For what values of A, B and n is the function (a) continuous, (b) differentiable, both at x = 0?
-
Use the limiting procedure of Eqn. (3.10) to differentiate:
(a) 2x3 + 4x + 3 (b) x− 2 and (c) f(x) = 5cos (3x).
-
Differentiate:
(a) x3ex (b)
(c) 
(d) arcsinh x (e)
(f) 3ln (1 + x2) -
Differentiate:
(a) sin (ln x) (b)
(c) 
(d) ln (ln x) -
Differentiate:
(a) xx (b) xcos x (c)
(d) 
-
Differentiate (a) y = ax and (b) y = log ax. For y = ax, compare your result with that obtained from the limiting procedure of Example 3.6 for a = 2.
-
Find
when x = t(1 + t2) and 
. Express your answer in terms of x and y. -
Neglecting air resistance, the path of a projectile moving under gravity is given by
where x is the horizontal distance travelled, y is the height, t is the time, and u and w are constants. Calculate the angle of flight and the rate of loss of kinetic energy at time t.
-
Find the points where the tangent to the curve
at (x, y) = (1, 1) intercept the x and y axes, respectively.
-
Find the value of
at the point (x, y) = (2, −1), where
and the equation of the normal to the curve at this point.
-
How many derivatives of the function
exist at x = 0?
-
Find general formulas for f(n)(x) where (a) f(x) = sinh 2x, (b) f(x) = ln x.
-
Show that
-
Prove the Leibnitz formula:
where
are the binomial coefficients.- Hence evaluate the fourth derivative of x2ln x.
-
(a) Find the stationary points of the function f(x) = x2exp ( − x2) and identify them as maxima or minima. (b) Sketch the resulting curve.
-
Locate the maxima and minima of f(x) = e− xsin x.
-
Make a sketch of the function y(x) = x3/(x2 − x − 2) showing clearly its main features.
-
Find the stationary points of the function
and state their nature. Locate any points of discontinuity and evaluate the limits of f(x) as x → ±∞. Sketch the form of f(x).
-
Use a graphical method to find approximate values for the real solutions of the equation x3 − 4x2 + x + 4 = 0.
-
Sketch the function f(x) = cos (10πx)exp ( − x2).
-
The normal at a point P(x1, y1) on an ellipse of eccentricity e and centre at the origin, as shown in Figure 3.9, meets the major axis at a pointA. If F is a focus, show that AF = ePF.
-
The tangent at a point P on the parabola y2 = 4ax meets the directrix at a point Q. The straight line through Q parallel to the axis of the parabola meets the normal at P at a point R. Show that the locus of R is a parabola and find its vertex.