James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.1 Derivatives of Polynomials and Exponential Functions 175
The Power Rule enables us to find tangent lines without having to resort to the definition
of a derivative. It also enables us to find normal lines. The normal line to a curve
C at a point P is the line through P that is perpendicular to the tangent line at P. (In the
study of optics, one needs to consider the angle between a light ray and the normal line
to a lens.)
Example 3 Find equations of the tangent line and normal line to the curve y − xsx
at the point s1, 1d. Illustrate by graphing the curve and these lines.
SOLUtion The derivative of f sxd − xsx − xx 1y2 − x 3y2 is
3
f 9sxd − 3 2 x s3y2d21 − 3 2 x 1y2 − 3 2 sx
tangent
normal
_1 3
_1
So the slope of the tangent line at (1, 1) is f 9s1d − 3 2 . Therefore an equation of the
tangent line is
y 2 1 − 3 2 sx 2 1d or y − 3 2 x 2 1 2
The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal
of 3 2 , that is, 22 3 . Thus an equation of the normal line is
FIGURE 4
y − xsx
y 2 1 − 2 2 3 sx 2 1d or y − 22 3 x 1 5 3
We graph the curve and its tangent line and normal line in Figure 4.
■
New Derivatives from Old
When new functions are formed from old functions by addition, subtraction, or multiplication
by a constant, their derivatives can be calculated in terms of derivatives of the old
func tions. In particular, the following formula says that the derivative of a constant times
a function is the constant times the derivative of the function.
Geometric Interpretation
of the Constant Multiple Rule
y
0
y=2ƒ
y=ƒ
Multiplying by c − 2 stretches the
graph vertically by a factor of 2. All
the rises have been doubled but the
runs stay the same. So the slopes are
doubled too.
x
The Constant Multiple Rule If c is a constant and f is a differentiable function,
then
d
dx fcf sxdg − c d dx f sxd
ProoF Let tsxd − cf sxd. Then
tsx 1 hd 2 tsxd
t9sxd − lim
− lim
h l 0 h
h l 0
− lim
h l 0
cF
− c lim
h l 0
f sx 1 hd 2 f sxd
h
f sx 1 hd 2 f sxd
h
G
cf sx 1 hd 2 cf sxd
h
(by Limit Law 3)
− cf 9sxd ■
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