Second and Higher Derivatives

Hitherto we have considered only the first derivative f(x) of a function y = f{x); now let us introduce the concept of second derivative (short for second-order derivative),, and derivatives of even higher orders. These will enable us to develop alternative criteria for locating the relative extrema of a function.

Derivative of a Derivative

Since the first derivative f'{x) is itself a function ofx, it, too, should be differentiate with respect to x, provided that it is continuous and smooth. The result of this differentiation, known as the second derivative of the function is denoted by where the double prime indicates that f(x) has been differentiated with respcct to x twice, and where the expression (r) following the double prime suggests that the second derivative is again a function of r or where the notation stems from the consideration that the second derivative d fdy\

means, in fact, — —- J; hence, the dL (read: "¿/-two") in the numerator dx \dx J

and dx2 (read; "cir squared1;) in the denominator of this symbol.

If the second derivative f"(x) exists for all .rvalues in the domain, the function fix) is said to be twice differentiable; if, in addition, f"(x) is continuous, the function /(x) is said to be twice continuously differentiate. Just as the notation / e C{{] or / e C is often used to indicate that the function/is continuously differentiate, an analogous notation

can be used to signify that/is twice continuously differentiate.

djy dx2

As a function of x the second derivative can be differentiated with respect to x again to produce a third derivative, which in turn can be the source of a fourth derivative, and so on ad infinitum, as long as the differentiability condition is met. These higher-order derivatives are symbolized along the same line as the second derivative:

/"'(■*). • • • > V) [with superscripts enclosed in ( )]

or dn 4 dn The last of these can also be written as -r—y* where the

. , _ part serves as an operator dxn dx" y symbol instructing us to take the nth derivative of (some function) with respect to x.

Almost all the specific functions we shall be working with possess continuous derivatives up to any order we desire; i.e., they are continuously differentiate any number of times. Whenever a general function is used, such as f(x), we always assume that it has derivatives up to any order we need.

Example 1

Find the first through the fifth derivatives of the function y=f(x) = 4x4-x3-M7x2 + 3x-1 The desired derivatives are as follows:

ff(x) = 16x? - 3x2 + 34* + 3 f'(x) = 48x2 - 6x + 34 r(x) = 96x^6 fiA)(x) = 96 f[5)(x) = 0

In this particular (polynomial) example, we note that each successive derivative function emerges as a lower-order polynomial—from cubic to quadratic, to linear, to constant We note also that the fifth derivative, being the derivative of a constant, is equal to zero for all values of x; we could therefore have written it as f^(x) - 0 as well. The equation fW(x) = 0 should be carefully distinguished from the equation f= 0 (zero at only). Also, understand that the statement f{h){x) = 0 does not mean that the fifth derivative does not exist; it indeed exists, and has the value zero,

Example 2

Find the first four derivatives of the rational function x

These derivatives can be found either by use of the quotient rule, or, after rewriting the function as y = x(1 + x)~\ by the product rule:

9"(x) = -2( 1+X)-3 gw(x) = 6(1 + x)"4 g(4)(x) = - 24(1+x)"5

In this case, repeated derivation evidently does not tend to simplify the subsequent derivative expressions.

Note that, like the primitive function g(x)< all the successive derivatives obtained are themselves functions ofx. Given specific values of v, however, these derivative functions will then take specific values. When x = 2, for instance, the second derivative in Example 2 can be evaluated as g"{2) = -2(3) 3 = ^

and similarly for other values ofx. It is of the utmost importance to realize that to evaluate this second derivative g"(,v) at x = 2, as we did, wc must first obtain gff(x) from g'{x) and then substitute x —2 into the equation for g"(x). It is incorrect to substitute * = 2 into g(x) or gr(x) prior to the differentiation process leading to g"(x).

Interpretation of the Second Derivative

The derivative function fix) measures the rate of change of the function/ By the same token, the second-derivative function /" is the measure of the rate of change of the first derivative /'; in other words, the second derivative measures the rate of change of the rate of change of the original function/ To put it differently, with a given infinitesimal increase in the independent variable x from a point x = jcq, f(xo)>0] i i .i j r i r j , (increase

/v/ v /\ r means that the value of the function tends to { . / (xn) < 0 J [ dccrcuse whereas, with regard to the second derivative, f,l(xb) 1 i j , i I increase

• , _ } means that the slope of the curve tends to { , / (xo) < 0 j [ decrease

Thus a positive first derivative coupled with a positive second derivative at x ~ x0 implies that the slope of the curve at that point is positive and increasing, tn other words, the value of the Junction is increasing at an increasing rate. Likewise, a positive first derivative with a negative second derivative indicates that the slope of the curve is positive but decreasing—the value of the function is increasing at a decreasing rate. The case of a negative first derivative can be interpreted analogously, but a warning is in order in this case: When ff(xa) < 0 and /" (*o) > 0, the slope of the curve is negative and increasing, but this does not mean that the slope is changing, say, from (-10) to (-11); on the contrary, the change should be from (-11), a smaller number, to (-10), a larger number. In other words, the negative slope must tend to be less steep as x increases. Lastly, when /'(*o) < 0 and ffT(xo) < 0, the slope of the curve must be negative and decreasing. This refers to a negative slope that tends to become steeper as x increases.

All of this can be further clarified with a graphical explanation. Figure 9.5a illustrates a function with /"(x) < 0 throughout. Since the slope must steadily decrease as x increases on the graph, we will, when wc move from left to right, pass through a point A with a positive slope, then a point B with zero slope, and then a point C with a negative slope. It may happen, of course, that a function with f"{x) < 0 is characterized by f(x) > 0 everywhere, and thus plots only as the rising portion of an inverse U-shaped curve, or, with ff(x) < 0 everywhere, plots only as the declining portion of that curve.

The opposite case of a function with /"(x) > 0 throughout is illustrated in Fig. 9.5h. Here, as we pass through points D to E to F, the slope steadily increases and changes from

x x negative to zero to positive. Again, we add that a function characterized by /"(*) > 0 throughout may, depending on the first-derivative specification, plot only as the declining or the rising portion of a U-shaped curve.

From Fig. 9.5, it is evident that the second derivative fff(x) relates to the curvature of a graph; it determines how the curve tends to bend itself. To describe the two types of differing curvatures discussed, we refer to the one in Fig. 9.5a as strictly concave, and the one in Fig. 9.56 as strictly convex. And, understandably a function whose graph is strictly concave (strictly convex) is called a strictly concave (strictly convex) function. The precise geometric characterization of a strictly concave function is as follows. If we pick any pair of points M and Won its curve and join them by a straight line, the line segment AW must lie entirely below the curve, except at points M and N* The characterization of a strictly convex function can be obtained by substituting the word above for the word below in the last statement. Try this out in Fig. 9,5, If the characterizing condition is relaxed somewhat, so that the line segment MN is allowed to lie either below the curve, or along (coinciding with) the curve, then we will be describing instead a concave function, without the adverb strictly. Similarly, if the line segment AW either lies above, or lies along the curve, then the function is convex, again without the adverb strictly. Note that, since the tine segment AW may coincide with a (nonstrictly) concave or convex curve, the latter may very well contain a linear segment. In contrast, a strictly concave or convex curve can never contain a linear segment anywhere. It follows that while a strictly concave (convex) function is automatically a concave (convex) function, the converse is not true.*

From our earlier discussion of the second derivative, we may now infer that if the second derivative /"(*) is negative for allx, then the primitive function f(x) must be a strictly concave function. Similarly, f(x) must be strictly convex, if fix) is positive for all x. Despite this, it is not valid to reverse this inference and say that, if f(x) is strictly concave (strictly convex), then /"(*) must be negative (positive) for all x. This is because, in certain exceptional cases, the second derivative may have a zero value at a stationary point on such a curve. An example of this can be found in the function y — f(x) = xA, which plots as a strictly convex curve, but whose derivatives

t We shall discuss these concepts further in Sec. 11.5.

indicate that, at the stationary point where x — 0, the value of the second derivative is /"(0) = 0. Note, however, that at any other point, with x ^ 0, the second derivative of this function does have the (expected) positive sign. Aside from the possibility of a zero value at a stationary point, therefore, the second derivative of a strictly concave or convex function may be expected in general to adhere to a single algebraic sign.

For other types of function, the second derivative may take both positive and negative values, depending on the value of x. In Fig. 93a and b, for instance, both f(x) and g{x) undergo a sign change in the second derivative at their respective inflection points J and K. According to Fig. 9.3a', the slope of f(x)—that is, the value of fff(x)—changes from negative to positive at x = y; the exact opposite occurs with the slope of#V)—that is, the value of gf{x)—on the basis of Fig. 9.W. Translated into curvature terms, this means that the graph of f(x) turns from strictly concave to strictly convex at point./, whereas the graph of g(x) has the reverse change at point K. Consequently, instead of characterizing an inflection point as a point where the first derivative reaches an extreme value, wc may alternatively characterize it as a point where the function undergoes a change in curvature or a change in the sign of its second derivative.

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