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MATH 140
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Ewa Duma
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Lecture 13

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Mathematics & Statistics (Sci)

MATH 140

Ewa Duma

Summer

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f▯x ▯ h▯ ▯ f▯x▯ c ▯ c
f▯▯x▯ ▯ lim ▯ lim ▯ lim 0 ▯ 0
hl0 h hl0 h hl0
0 x
In Leibniz notation, we write this rule as follows.
FIGURE 1
The graph of ƒ=c is the
line y=c, so fª(x)=0. DERIVATIVE OF A CONSTANT FUNCTION MATH140 - Lecture 13 Notes
d
▯c▯ ▯ 0
dx
Derivatives of Polynomials and Exponential Functions:
Derivative of a constant function: d/dx (c) = 0
POWER FUNCTIONS
y n
We next look at the functions f▯x▯ ▯ x , where n is a positive integer. If n ▯ 1, the graph
of f▯x▯ ▯ x is the line y ▯ x, which has slope 1. (See Figure 2.) So
y=x
1766 |||||CHAPTER 3 DIFFERENTIATION RULESS
Power Functions:
slope=1 d
1 V EXAMPLE 3 Find equations dx the tangent line and normal line to the curve y ▯sx x
V EXAMPLE 3 For functions f(x) = x^n, where n is a positive integer. If n = 1, the graph of f(x) = x is the
0 at the point ▯1, 1▯. Illustrate by graphing the curve and these lines.
x line y = x, which has a slope1▯2 1. 3▯2refore, the derivative of x is equal to 1.
(You canSOLUTION The derivative of f▯x▯ ▯sxx ▯ xxn of ▯ xeriistive.) We have already
FIGURE 2 investigated the casesExamples:d n ▯ 3. In 3ac▯3▯2▯▯1e3ti1▯22.3 (Exercises 17 and 18) we
The graph of ƒ=x 3s the found that f▯▯x▯ ▯ 2x ▯ x2 ▯▯ 2ssxx
line y=x, so fª(x)=1. d d 3
tangent 2 So the slope of the▯x ▯ ▯ 2xline at (1, ▯x ▯ ▯ 3x2▯ ▯ . Therefore an equation of the tan-
6 030102 tangent gent line is dx dx
3 33 11
y ▯ 1 ▯ ▯2 ▯ 1▯ orr y ▯ 22▯ 22
normall For n ▯ 4 we ﬁnd the derivative of f▯x▯ ▯ x as follows:
The normal line is perpendicular to the tangent line, so its slope is the negative recipro-er, then we call
_1 3 The no3mal line i22perpendicular to the tangent line, so its slope is the negative recipro-
_1 3 cal of2, that is,33 Thf▯x ▯ h▯ ▯ f▯x▯of the no▯x ▯ h▯ ▯ xs 4
f▯▯x▯ ▯ lim ▯ lim
hl0y ▯ 1 ▯ ▯ ▯x ▯ 1▯ hl0orr y ▯ ▯ x ▯ 5
_1 The evide4ce lie3 in the2 2llowing3formu4a: 4 3
x ▯ 4x h ▯ 6x h ▯ 4xh ▯ h ▯ x
FIGURE 44 We graph the curvehl0d its tangent line hnd normal line in Figure 4. MM
X^n - a^n = (x-a)(x^(n-1)+x^(n-2)a + … + xa^(n-2) + a^(n-1))
3 2 2 3 4
NEW DERIVATIVES▯ lim OLD h ▯ 6x h ▯ 4xh ▯ h
hl0 h
When new functions are formed from old functions by addition, subtraction, or multipli-
▯ lim ▯4x ▯ 6x h ▯ 4xh ▯ h ▯ ▯ 4x 3
cation by a constahl0 their derivatives can be calculated in terms of derivatives of the old
functions. In particular, the following formula says that the derivative of a constant timesn is the
Thus a function is constant times the derivative of the function.ion.
3 d ▯x ▯ ▯ 4x 3
NNGGEOMETRIC INTERPRETATION OF dx
TTHE CONSTANT MULTIPLE RULE THE CONSTANDeﬁnition: The constant multiple rule says that if c is a constant and f is a differentiable
tion, then function, then:
y
dd d 173
yy=2ƒ ▯cf▯x▯▯ ▯ c f▯x▯
dx ddx d/dx [cf(x)] = c(d/dx)⋅f(x)
yy=ƒ
Assuming that g(x) = cf(x), we can show the following:
PROOF LLet t▯x▯ ▯ cf▯x▯. Then
00 x
t▯x ▯ h▯ ▯ t▯x▯ cf▯x ▯ h▯ ▯ cf▯x▯
t▯▯x▯ ▯ lim ▯▯ lim
MMultiplying by c ▯ 2 stretches the graph verti- hhl00 h hhl00 h
ccally by a factor of 2. All the rises have been
doubled but the runs stay the same. So the f▯x ▯ h▯ ▯ f▯x▯

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