examples a: derivatives involving algebraic functions
TRANSCRIPT
Examples A: Derivatives Involving Algebraic Functions
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
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:
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Example (2)
28)7(4)2())1(
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)())(()(
))(()(
:
)2(
7)2(,1)2(&4)1(
))(()(:
hg
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Soluion
fFind
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Examples B: Derivatives Involving Trigonometric Functions
Basic Formulas
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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
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Examples (2)
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1920
20
20tansec
sec.3
20sec
tan.2
20sin
cos.1
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xy
Examples (3)
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cossin20
)(sinsin.4
tansecsec20
)(secsec.3
sectan20
)(tantan.2
)sin(cos20
)(coscos.1
19
2020
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19
2020
Examples (4)
21
2
22
78
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8
11:
61
cot
cscsin5cos561
cot
)!(6
1cot
sin5.2
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)!(5sin.1
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sec
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])(tan432[10
]5sec)(tan3
242[
])(tan432[10
])(tan432[
)tan432(
)tan432(.4
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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
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Examples (7)
))18(tan(sin.3
)8(tansin.2
)8sin(tan.1
359
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8sec
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)8sin(tan.1
x
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8sec
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)]8[sin(tan9
)]8[sin(tan
)8(tansin.2
x
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)18(7
)18(sec
])18tan(cos[
])18tan(sin[9
])18tan(sin[
])18tan([sin.3
x
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xy