Chapter 4 | Applications of Derivatives
359
divide both sides of Equation 4.2 by dx , which yields
dy dx =
(4.3)
f ′( x ). This is the familiar expression we have used to denote a derivative. Equation 4.2 is known as the differential form of Equation 4.3 . Example 4.8 Computing differentials
For each of the following functions, find dy and evaluate when x =3 and dx =0.1. a. y = x 2 +2 x b. y =cos x Solution The key step is calculating the derivative. When we have that, we can obtain dy directly. a. Since f ( x ) = x 2 +2 x , we know f ′( x ) =2 x +2, and therefore dy = (2 x +2) dx .
When x =3 and dx =0.1,
dy = (2 · 3 + 2)(0.1) = 0.8.
b. Since f ( x ) =cos x , f ′( x ) =−sin( x ). This gives us
dy =−sin xdx .
When x =3 and dx =0.1,
dy = −sin(3)(0.1) = −0.1sin(3).
2 , find dy .
4.8
For y = e x
We now connect differentials to linear approximations. Differentials can be used to estimate the change in the value of a function resulting from a small change in input values. Consider a function f that is differentiable at point a . Suppose the input x changes by a small amount. We are interested in how much the output y changes. If x changes from a to a + dx , then the change in x is dx (also denoted Δ x ), and the change in y is given by Δ y = f ( a + dx )− f ( a ). Instead of calculating the exact change in y , however, it is often easier to approximate the change in y by using a linear approximation. For x near a , f ( x ) can be approximated by the linear approximation L ( x ) = f ( a )+ f ′( a )( x − a ). Therefore, if dx is small, f ( a + dx ) ≈ L ( a + dx ) = f ( a )+ f ′( a )( a + dx − a ).
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