11DIFFERENTIATION - Department of Mathematics

The Differential
EXAMPLE 3
SOLUTION ✔
11.7 DIFFERENTIALS 773
Observe that near the point of tangency P, the tangent line T is close to the
graph of f. Therefore, if x is small, then dy is a good approximation of y.
We can find an expression for dy as follows: Notice that the slope of T is
given by
dy
(Rise divided by run)
x
However, the slope of T is given by f(x). Therefore, we have
dy
f(x)
x
or dy f(x)x. Thus, we have the approximation
y dy f(x)x
in terms of the derivative of f at x. The quantity dy is called the differential
of y.
Let y f(x) define a differentiable function of x. Then,
1. The differential dx of the independent variable x is dx x.
2. The differential dy of the dependent variable y is
REMARKS
dy f(x)x f(x)dx (11)
1. For the independent variable x: There is no difference between x and
dx—both measure the change in x from x to x x.
2. For the dependent variable y: y measures the actual change in y as x
changes from x to x x, whereas dy measures the approximate change
in y corresponding to the same change in x.
3. The differential dy depends on both x and dx, but for fixed x, dy is a linear
function of dx.
Let y x3 .
a. Find the differential dy of y.
b. Use dy to approximate y when x changes from 2 to 2.01.
c. Use dy to approximate y when x changes from 2 to 1.98.
d. Compare the results of part (b) with those of Example 2.
a. Let f(x) x 3 . Then,
dy f(x) dx 3x 2 dx