114 Chapter 4Differentiation
For example,
n 1 = 1 3: f
(3)(x) 1 = 1 f′′′(x) 1 = 1 −a
31 cos 1 ax
n 1 = 1 4: f
(4)(x) 1 = 1 (−1)
2a
41 sin 1 ax 1 = 1 a
41 sin 1 ax
0 Exercises 75 –77
The first derivativef′(x)of a functionf(x)is the rate of change of the function, or
the slope of its graph at point x. The second derivativef′′(x)is the rate of change of
slope, and is related to the curvature at x.
4.10 Stationary points
Consider the cubic (Figure 4.9)
y 1 = 1 x(x 1 − 1 3)
2The function goes to±∞asx 1 → 1 ±∞, but the graph
shows that the function has a local maximumat
point A, atx 1 = 11 , where its value is greater than at
all neighbouring points. The function also has a
(local) minimumat point B, atx 1 = 13 , where its
value is smaller than at all neighbouring points.
Points of maximum and minimum value are called
turning points.
The determination of the maximum and mini-
mum values of a function is of importance in the
physical sciences because, for example, (i) equations of motion are often formulated
as ‘variation principles’, whereby solutions are obtained as maxima or minima of
some variational function (see Example 4.25), (ii) the fitting of a theoretical curve to
a set of experimental points can be expressed in terms of a ‘minimum deviation’
principle, as in the method of least squares discussed in Chapter 21.
Consider the curve in the neighbourhood of the maximum at point A in Figure 4.9.
To the left of A the gradient is positive and the value of the function is increasing. To
the right of A the gradient is negative and yis decreasing. At the point A itself the
function has zero gradient (the tangent to the curve is horizontal), and the rate of
change of ywith respect to xis zero. The point is called a stationary point, and the
value of the function at the point is called a stationary value. Similar considerations
apply for the minimum at B, and the general condition for a stationary point is that
the first derivative of the function be zero:
at a stationary point (4.20)
To distinguish between maximum and minimum values, it is necessary to consider
the second derivative. On moving through the maximum at A from left to right, the
dy
dx
= 0
1
2
3
4
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Figure 4.9