11DIFFERENTIATION - Department of Mathematics

752 11 DIFFERENTIATION
FIGURE 11.13
The graph of the function y x 2/3
EXAMPLE 3
SOLUTION ✔
–4 –2
4
2
y
2 4
The common domain of the functions f, f , and f is the set of all real
numbers except x 0. The domain of y x 2/3 is the set of all real numbers.
The graph of the function y x 2/3 appears in Figure 11.13.
REMARK Always simplify an expression before differentiating it to obtain
the next order derivative.
Find the second derivative of the function y (2x 2 3) 3/2 .
We have, using the general power rule,
y 3
2 (2x2 3) 1/2 (4x) 6x(2x 2 3) 1/2
Next, using the product rule and then the chain rule, we find
y(6x) d
dx (2x2 3) 1/2 d
dx (6x) (2x2 3) 1/2
(6x)1 (2x 2 2 3) 1/2 (4x) 6(2x2 3) 1/2
12x2 (2x2 3) 1/2 6(2x2 3) 1/2
APPLICATIONS
x
6(2x2 3) 1/2 [2x2 (2x2 3)]
6(4x2 3)
2x2 3
Just as the derivative of a function f at a point x measures the rate of change
of the function f at that point, the second derivative of f (the derivative of f)
measures the rate of change of the derivative f of the function f. The third
derivative of the function f, f , measures the rate of change of f , and so on.
In Chapter 12 we will discuss applications involving the geometric interpretation
of the second derivative of a function. The following example gives an
interpretation of the second derivative in a familiar role.