Calculus and Analytical Geometry Lecture # 8 MTH 104.
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Transcript of Calculus and Analytical Geometry Lecture # 8 MTH 104.
![Page 1: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/1.jpg)
Calculus and Analytical Geometry
Lecture # 8
MTH 104
![Page 2: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/2.jpg)
Techniques of differentiation
1. Constant Function Rule:The derivative of a constant function is zero. y = f(x) = cwhere c is a constant
.0
dxdc
dxxdf
dxdy
Examples
,0)1(
dxd ,0
)5( dxd .0
)2( dx
d
![Page 3: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/3.jpg)
Techniques of differentiation
nxxfy )(
xxxdxd
22 122
1 nn nxxdxd
2. Power Rule:Let , where the dependant
variable x is raised to a constant value, the power n, then
5xdxd
45x
Examples
![Page 4: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/4.jpg)
Techniques of differentiation
2
1
xdxd
xxx
21
21
21
2
11
2
1
7xdxd 817 77 xx
78 8xxdxd
![Page 5: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/5.jpg)
3. Constant Multiplied by a Function Rule:Let y be equal to the product of a constant c and some function f(x), such that y = cf(x) then
Techniques of differentiation
dxxdf
cdxxcfd
dxdy )())((
dxxd )4( 3
dxxd )(4 3
Examples
213 1234 xx
![Page 6: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/6.jpg)
Techniques of differentiation
1112
1212
x
xdxd
xdxd
1
2
1
x
xdxd
xdxd
xdxd
![Page 7: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/7.jpg)
Techniques of differentiation
xgdxd
xfdxd
xgxfdxd
dxdy
962 xxdxd 962 x
dxd
xdxd
4. Sum (Difference) Rule:Let y be the sum (difference) of two functions (differentiable) f(x) and g(x).
y = f(x) + g(x),
then
Examples
)9(62 195 xx105 912 xx
![Page 8: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/8.jpg)
Techniques of differentiation
x
x
x
xdxd
dxd
xdxd
1
21
-2
dxd
2-0
2121
2
1
2
1
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Techniques of differentiation.9523 38 xxxy
9523 38 xxxdxd
dxdy
9523 38
dxd
xdxd
xdxd
xdxd
Example Find dy/dx if
solution
0)1(53283 27 xx
5624 27 xx
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Techniques of differentiation
433 xxy
33
43
2
3
x
xxdxd
dxdy
Example At what points, if any does the graph of
have a horizontal tangent line? solution
1
1
01
033 0
2
2
2
x
x
x
xdxdy
Slope of horizontal line is zero that is dy/dx=0
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Techniques of differentiation
4. Product Rule:Let y = f(x).g(x), where f(x) and g(x) are two
differentiable functions of the variable x. Then
xfdxd
xgxgdxd
xfdxdy
xgxfdd
dxdy
![Page 12: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/12.jpg)
Techniques of differentiation
xxxy 32 714
xxxdxd
dxdy 32 714
147714 2332 xdxd
xxxxdxd
x
Example Find dy/dx, if
solution
xxxxx 8712114 322
19140
85612148424
24224
xx
xxxxx
![Page 13: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/13.jpg)
Techniques of differentiation
5. Quotient Rule:Let y = f(x)/g(x), where f(x) and g(x) are two differentiable functions of the variable x and g(x) ≠ 0. Then
2
xg
xgdxd
xfxfdxd
xg
dxdy
xgxf
dxd
dxdy
![Page 14: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/14.jpg)
Techniques of differentiation
514
2
xx
y
514
2xx
dxd
dxdy
22
22
5
5)14(145
x
xdxd
xxdxd
x
Example Find dy/dx if
solution Derivative of numerator
Derivative of denominator
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Techniques of differentiation
22
2
52)14(45
xxxx
22
22
528204
xxxx
22
2
52024
xxx
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Higher order derivatives
nfffff 4,,,If y=f(x) then
xfdxd
dxyd
y
xfdxd
xfdxd
dxd
dxyd
y
xfdxd
dxdy
y
3
3
3
3
2
2
2
2
![Page 17: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/17.jpg)
Higher order derivatives
A general nth order derivative
xf
dxyd
xfxfdxd
dxyd
n
n
n
n
n
n
n
n
and
![Page 18: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/18.jpg)
Example
constants. are
,, where if Find 24 cbacbxaxyy
cbxaxdxd
dxdy
y 24
cdxd
bxdxd
axdxd 24
Solution
bxax 24 3 bxax
dxd
dxyd
y 24 3
2
2
First Orderderivat
ive
Second orderderivative
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)2()4(2
2
3
2
2
2
2
bxdxd
axdxd
dxyd
y
bax 212 2
ax
baxdxd
dxyd
y
12
212 2
3
3
3
3
Third order derivative
![Page 20: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/20.jpg)
Example Find 25
1x
2
2
46 where xxydx
yd
xx
xx
xxdxd
dxdy
830
2456
46
4
4
25
xxdxd
dxyd
830 4
2
2
Solution
8120 3 x
1128)1(1201
2
2
x
dx
yd
![Page 21: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/21.jpg)
Derivative of trigonometric functions
xxdxd
cossin .1
xcoxdxd
sin .2
xxdxd
2sectan .3
xxxdxd
tansecsec .5
xxxdxd
cotcsccsc .4
xxdxd
2csccot .6
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Example . find cossin2
2
dxyd
xxy
xxdxd
dxdy
cossin
xdxd
xxdxd
x sincoscossin
Solution
)(coscos)sin(sin xxxx
xx 22 cossin
xdxd
xdxd
dxyd
22
2
2
cossin
xx
xxxx
cossin4
cossin2cossin2
![Page 23: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/23.jpg)
Example
xyy
xxy
cos2
osolution t a is sin that Show
1 cos2 xyy
xxxy r.t w.sin times twoatingDifferenti
solution
xxx
xxdxd
dxdy
y
sincos
sin
xdxd
xxdxd
xxxdxd
y sincossincos
![Page 24: Calculus and Analytical Geometry Lecture # 8 MTH 104.](https://reader036.fdocuments.net/reader036/viewer/2022081501/5697bfa91a28abf838c99bc9/html5/thumbnails/24.jpg)
xxx
xxxxy
cos2sin
coscossin
yy and
xxxxxx cos2sincos2sin
Substituting the valuse of into (1)
xx cos2cos2 L.H.S=R.H.S
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Example Given that xxxf tansin)( 2 24
fshow that
xxxf tansin)( 2
xxxf tansin)( 2xx 22 secsin )(xf ))(cos(sin2 tan xxx
xx
22
cos
1sin
x2tan x2sin2
4f
4tan2
4
sin2 2
12
2
12
2
1
))(cos(sin2
cos
sin xx
x
x