calculo-clases pucp
DESCRIPTION
Clases de matematica pucpTRANSCRIPT
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Capítulo 4
La Derivada
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La Derivada
)()()(
lim)(
:)()2
)()()(
lim)(
:)1
0
'
0
'
existesih
xfhxfxf
xfdederivadaLa
existesih
afhafaf
aenfdederivadaLa
realfunciónunafSea
h
h
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Nota
ax
afxf
h
afhafaf
existeafSi
axh
)()(lim
)()(lim)(
:)(
0
'
'
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Recta Tangente
Si f ’(a) existe:
La pendiente de la recta tangente a y = f(x)
en el punto (a,f(a)) es mT = f ’(a).
La recta tangente a y = f(x)
en el punto (a,f(a)) es y = f ’(a).(x-a)+f(a).
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Nota
La recta tangente a y = f(x)en el punto (a,f(a)) es x = a.
esh
afhafy
esh
afhafSi
h
h
)()(lim
)()(lim
0
0
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Recta Normal
La recta normal a una gráfica en un punto dado
es la recta perpendicular a la recta tangente en
ese punto.
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Velocidad
Si una partícula P se mueve en una recta
y e(t) es la coordenada de P en el instante t,
la velocidad de P en el instante t es: v(t) = e’(t)
y la aceleración de P en el instante t es: a(t) = v’(t)
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Función Derivable
Sea f una función real
1) f es derivable en a,
si existe f ’(a).
2) f es derivable en el intervalo abierto I,
si existe f ’(x) para todo x I.
3) f es derivable,
si existe f ’(x) para todo x Dom(f).
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Teorema
Si f(x) = mx+b f ’(x) = m.
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Corolario
Si f(x) = c f ’(x) = 0.
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Notación
dx
dyyDy
xfySidx
xdfxfDaf
aenfdederivadaLadx
xdfxfDxf
xfdederivadaLa
x
axaxx
x
'
:)()3
|)(
|)]([)(
:)2
)()]([)(
:)()1
'
'
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Derivadas Laterales
1) La derivada por la derecha de f en a:
2) La derivada por la izquierda de f en a:
)()()(
lim)()(
lim)(0
' existesiax
afxf
h
afhafaf
axh
)()()(
lim)()(
lim)(0
' existesiax
afxf
h
afhafaf
axh
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Teorema
))()(())(( ''' mafmafmaf
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Teorema
Si f es derivable en a f es continua en a
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Teorema
1' )(
)()(
n
n
nxxf
ZnxxfSi
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Teorema
0,2
1)(
0,)(
'
xx
xf
xxxfSi
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Teorema
)()()cos()()2
)cos()()()()1'
'
xsenxfxxf
xxfxsenxf
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Teorema
Si f es derivable en x
(c.f)’(x) = c.f ’(x)
( Dx[c.f(x)] = c.Dx[f(x)] )
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Teorema
Si f y g son derivables en x
(f+g)’(x) = f ’(x)+g’(x)
( Dx[f(x)+g(x)] = Dx[f(x)]+Dx[g(x)] )
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Corolario
Si f1,…,fn son derivables en x
(f1+…+fn)’(x) = f1’(x) +…+fn’(x)
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Teorema
Si f y g son derivables en x
(f.g)’(x) = f ’(x).g(x)+f(x).g’(x)
( Dx[f(x).g(x)] = Dx[f(x)].g(x)+f(x).Dx[g(x)] )
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Teorema
Si f y g son derivables en x g(x) 0
2
'''
))((
)().()().()()(
xg
xgxfxgxfx
g
f
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Teorema
)()(
)()(1' existesinxxf
ZnxxfSin
n
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Teorema
1) Dx[tan(x)] = sec2(x)
2) Dx[cot(x)] = -csc2(x)
3) Dx[sec(x)] = sec (x).tan(x)
4) Dx[csc(x)] = -csc(x).cot(x)
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Teorema(Derivada de la función compuesta)
Si g es derivable en x y f es derivable en g(x) (f o g)’(x) = f ’(g(x)).g’(x)
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Nota (Regla de la cadena)
dx
du
du
dy
dx
dy.
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Teorema
)()(
)()(1' existesinxxf
QnxxfSin
n
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Teorema
xaxfxxf
aaxfaxf
xxfxxf
exfexf
a
xx
xx
1.)ln(
1)()(log)()4
).ln()()()3
1)()ln()()2
)()()1
'
'
'
'
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Teorema
)()(
)()(1' existesinxxf
IRnxxfSin
n
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Teorema (Derivada de la función inversa)
Sea f una función montona y derivable en I =[a,b],
f ’(x) 0, x I.
))(
1(
)))((
1)()((
)(
1)()()(
1''1
''1
dx
dydy
dx
xffxf
xfyfxfySi
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Teorema
21
2
1
2
1
1
1)]([tan)3
1
1)]([cos)2
1
1)]([)1
xxD
xxD
xxsenD
x
x
x
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1||
1)]([csc)6
1||
1)]([sec)5
1
1)]([cot)4
2
1
2
1
21
xxxD
xxxD
xxD
x
x
x
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Derivación Implícita
),(
))((0)],([
0),(
yxGdx
dy
xfyyxFD
yxFSi
x
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Derivada Paramétrica
)(
)(
)(
)(
dt
dxdt
dy
dx
dy
tgy
tfxSi
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Derivadas de orden superior
)1(,))(()(
)()(')()1(
')1(
nxfxf
xfxfnn
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Nota
)())(())(()(
)())(())(()(
)()(
''''''')2()3(
''''')1()2(
')1(
xfxfxfxf
xfxfxfxf
xfxf
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Notación
n
nnx
n
axn
n
axnx
n
n
nnx
n
dx
ydyDyxfySi
dx
xfdxfDaf
dx
xfdxfDxf
][:)()3
|)(
|)]([)()2
)()]([)()1
)(
)(
)(
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Nota
dx
dx
xfdd
dx
xfd
xfDDxfD
xfxf
n
n
n
n
nxx
nx
nn
1
1
1
')1()(
)(
)()3
)]]([[)]([)2
))(()()1
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Nota
dt
dx
dtdx
ydd
dxdx
ydd
dx
yd
tgy
tfxSi
n
n
n
n
n
n
)(
)(
)(
)(
1
1
1
1
![Page 40: Calculo-clases pucp](https://reader036.vdocuments.co/reader036/viewer/2022081502/5695d30e1a28ab9b029cafb0/html5/thumbnails/40.jpg)
La diferencial
Sea y = f(x).
La diferencial de x: dx = x.
La diferencial de y: dy = f ’(x).dx
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Aproximación por diferenciales
dyyxyyyx
y
xSix
yy
xfxxfy
xfySi
x
'.'
:0
lim'
).()(
:)(
0
![Page 42: Calculo-clases pucp](https://reader036.vdocuments.co/reader036/viewer/2022081502/5695d30e1a28ab9b029cafb0/html5/thumbnails/42.jpg)
Aproximación lineal
)()).(()(
)()()(
:
)()(lim)(
'
'
'
afaxafxf
afax
afxf
axSiax
afxfaf
ax