examples a: derivatives involving algebraic functions
Post on 14-Jan-2016
279 Views
Preview:
TRANSCRIPT
The derivative of composite functionfor the case f(x) = gn(x)
Let:f(x) = gn(x)Then:f' (x) = ngn-1(x) . g'(x)
Example:Let f(x) = (3x8 - 5x + 3 )20
Then f(x) = 20 (3x8 - 5x + 3 )19 (24x7 - 5)
Examples (1)
36
7
5 9
9
7
5 9
9
5 99 59
5 99 599
9
45
32)(.3
32)(.2
1111)(.1
:
xx
xxxf
xxxf
xxxxxx
xxxf
functionsfollowingtheofeachofderivativetheFind
514
914
54
94
910
98
59
95
59
95
91
91
5
9
9
5
5
99
5
9
1
9
199)(
11
11)(.1
108
99
5 99 5
5 9
9 5
9
99
9
xxx
xxxxxxf
xxxxxxxx
xxx
xx
xx
xxf
514
59
59
59
5
271832745
43045332)(
4532
45
32)(.3
86
9316
252167
9
3167
9
36
7
5 9
9
xxxxxx
xxxxxxxf
xxxx
xx
xxxf
Example (2)
28)7(4)2())1(
)2()1()2(
)2())2(()2(
)())(()(
))(()(
:
)2(
7)2(,1)2(&4)1(
))(()(:
hg
hgf
hhgf
xhxhgxf
xhgxf
Soluion
fFind
hhg
xhgxfLet
Basic Formulas
xxyxy
xxyxy
xyxy
xyxy
xyxy
xyxy
cotcsccsc.5
tansecsec.5
csccot.4
sectan.3
sincos.2
cossin.1
2
2
General Formulas (Chain Rule)Let u=u(x)
uuuyuy
uuuyuy
uuyuy
uuyuy
uuyuy
uuyuy
cotcsccsc.5
tansecsec.5
csccot.4
sectan.3
sincos.2
cossin.1
2
2
Examples (1)
)!()1(csccos
sin
coscot;coscsccos
sincotcsccos
)sin(cot)csc(cos
cotcos.3
)sec(
)sectansec1(tansec)sec(
sec
tan.2
2sin82cos2
82sin22cos
2sin.1
2
2
2
2
2
2
78
78
8
timprovemenmuchnotxx
x
xxbecausexxx
xxxx
xxxxy
xxy
xxx
xxxxxxxxxy
xxx
xy
xxxx
xxxxy
xxy
Examples (2)
192020
20
19202
20
1920
20
20tansec
sec.3
20sec
tan.2
20sin
cos.1
xxxy
xy
xxy
xy
xxy
xy
Examples (3)
xxy
xxy
xxxy
xxy
xxy
xxy
xxy
xxy
cossin20
)(sinsin.4
tansecsec20
)(secsec.3
sectan20
)(tantan.2
)sin(cos20
)(coscos.1
19
2020
19
2020
219
2020
19
2020
Examples (4)
21
2
22
78
78
8
11:
61
cot
cscsin5cos561
cot
)!(6
1cot
sin5.2
5sin85cos5
85sin55cos
)!(5sin.1
xx
soandxx
Note
xx
xxxxx
xy
quotient
xx
xy
xxxx
xxxxy
productxxy
]tan
sec
3
402[
])(tan432[10
]5sec)(tan3
242[
])(tan432[10
])(tan432[
)tan432(
)tan432(.4
3 5
524
93
25
4523
15
93
25
103
25
1053
2
103 52
x
xx
xxy
xxx
xxy
xx
xx
xxy
Examples (6)
)18()]32cot()32csc([)]32[csc(5
7
)]32[csc()32(csc
)32(csc.2
)18()32(5
7)32cot()32csc(
)32csc(
)32(csc.1
3445
24
5
7445
7
5 47
35
245
745
74
5
74
5 74
xxxxxxxy
xxxx
xxy
xxxxxxxy
xx
xxy
top related