Derivada Regras Operatorias

Prévia do material em texto

Ca´lculo Diferencial e Integral I
Regras Operato´rias das Derivadas
Luiz C. M. de Aquino
aquino.luizclaudio@gmail.com
http://sites.google.com/site/lcmaquino
http://www.youtube.com/LCMAquino
Regras Operato´rias das Derivadas
Introduc¸a˜o
Tabela ba´sica de derivadas
f (x) f ′(x)
c 0
xn nxn−1
n
√
x
1
n
n
√
xn−1
ax ax ln a
loga x
1
x ln a
sen x cos x
cos x − sen x
Regras Operato´rias das Derivadas
Teorema
Sejam f e g func¸o˜es diferencia´veis em um mesmo dom´ınio D. Sa˜o
va´lidas as afirmac¸o˜es:
(i) [cf (x)]′ = cf ′(x), com c uma constante real qualquer;
(ii) [f (x) + g(x)]′ = f ′(x) + g ′(x);
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x);
(iv)
[
f (x)
g(x)
]′
= f
′(x)g(x)−f (x)g ′(x)
[g(x)]2
, com g(x) 6= 0 no dom´ınio D.
Observac¸a˜o
De (i) e (ii) segue que:
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x).
Regras Operato´rias das Derivadas
Teorema
Sejam f e g func¸o˜es diferencia´veis em um mesmo dom´ınio D. Sa˜o
va´lidas as afirmac¸o˜es:
(i) [cf (x)]′ = cf ′(x), com c uma constante real qualquer;
(ii) [f (x) + g(x)]′ = f ′(x) + g ′(x);
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x);
(iv)
[
f (x)
g(x)
]′
= f
′(x)g(x)−f (x)g ′(x)
[g(x)]2
, com g(x) 6= 0 no dom´ınio D.
Observac¸a˜o
De (i) e (ii) segue que:
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x).
Regras Operato´rias das Derivadas
(i) [cf (x)]′ = cf ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[cf (x)]′ = lim
h→0
cf (x + h)− cf (x)
h
= lim
h→0
c[f (x + h)− f (x)]
h
= c lim
h→0
f (x + h)− f (x)
h
= cf ′(x)
Regras Operato´rias das Derivadas
(i) [cf (x)]′ = cf ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[cf (x)]′ = lim
h→0
cf (x + h)− cf (x)
h
= lim
h→0
c[f (x + h)− f (x)]
h
= c lim
h→0
f (x + h)− f (x)
h
= cf ′(x)
Regras Operato´rias das Derivadas
(i) [cf (x)]′ = cf ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[cf (x)]′ = lim
h→0
cf (x + h)− cf (x)
h
= lim
h→0
c[f (x + h)− f (x)]
h
= c lim
h→0
f (x + h)− f (x)
h
= cf ′(x)
Regras Operato´rias das Derivadas
(i) [cf (x)]′ = cf ′(x)
Exemplo 1: Seja a func¸a˜o f (x) = 5x3.
f ′(x) =
(
5x3
)′
= 5
(
x3
)′
= 5
(
3x2
)
= 15x2
Regras Operato´rias das Derivadas
(i) [cf (x)]′ = cf ′(x)
Exemplo 1: Seja a func¸a˜o f (x) = 5x3.
f ′(x) =
(
5x3
)′
= 5
(
x3
)′
= 5
(
3x2
)
= 15x2
Regras Operato´rias das Derivadas
(i) [cf (x)]′ = cf ′(x)
Exemplo 1: Seja a func¸a˜o f (x) = 5x3.
f ′(x) =
(
5x3
)′
= 5
(
x3
)′
= 5
(
3x2
)
= 15x2
Regras Operato´rias das Derivadas
(i) [cf (x)]′ = cf ′(x)
Exemplo 1: Seja a func¸a˜o f (x) = 5x3.
f ′(x) =
(
5x3
)′
= 5
(
x3
)′
= 5
(
3x2
)
= 15x2
Regras Operato´rias das Derivadas
(ii) [f (x) + g(x)]′ = f ′(x) + g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x) + g(x)]′ = lim
h→0
[f (x + h) + g(x + h)]− [f (x) + g(x)]
h
= lim
h→0
[f (x + h)− f (x)] + [g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
+ lim
h→0
g(x + h)− g(x)
h
= f ′(x) + g ′(x)
Regras Operato´rias das Derivadas
(ii) [f (x) + g(x)]′ = f ′(x) + g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x) + g(x)]′ = lim
h→0
[f (x + h) + g(x + h)]− [f (x) + g(x)]
h
= lim
h→0
[f (x + h)− f (x)] + [g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
+ lim
h→0
g(x + h)− g(x)
h
= f ′(x) + g ′(x)
Regras Operato´rias das Derivadas
(ii) [f (x) + g(x)]′ = f ′(x) + g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x) + g(x)]′ = lim
h→0
[f (x + h) + g(x + h)]− [f (x) + g(x)]
h
= lim
h→0
[f (x + h)− f (x)] + [g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
+ lim
h→0
g(x + h)− g(x)
h
= f ′(x) + g ′(x)
Regras Operato´rias das Derivadas
(ii) [f (x) + g(x)]′ = f ′(x) + g ′(x)
Exemplo 2: Seja a func¸a˜o f (x) = 2x + cos x .
f ′(x) = (2x + cos x)′
= (2x)′ + (cos x)′
= 2x ln 2− sen x
Regras Operato´rias das Derivadas
(ii) [f (x) + g(x)]′ = f ′(x) + g ′(x)
Exemplo 2: Seja a func¸a˜o f (x) = 2x + cos x .
f ′(x) = (2x + cos x)′
= (2x)′ + (cos x)′
= 2x ln 2− sen x
Regras Operato´rias das Derivadas
(ii) [f (x) + g(x)]′ = f ′(x) + g ′(x)
Exemplo 2: Seja a func¸a˜o f (x) = 2x + cos x .
f ′(x) = (2x + cos x)′
= (2x)′ + (cos x)′
= 2x ln 2− sen x
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x)g(x)]′ = lim
h→0
f (x + h)g(x + h)− f (x)g(x)
h
= lim
h→0
f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h)
h
= lim
h→0
[f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)]
h
= lim
h→0
[f (x + h)− f (x)]g(x + h)
h
+ lim
h→0
f (x)[g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
lim
h→0
g(x + h) + f (x) lim
h→0
g(x + h)− g(x)
h
= f ′(x)g(x) + f (x)g ′(x)
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x)g(x)]′ = lim
h→0
f (x + h)g(x + h)− f (x)g(x)
h
= lim
h→0
f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h)
h
= lim
h→0
[f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)]
h
= lim
h→0
[f (x + h)− f (x)]g(x + h)
h
+ lim
h→0
f (x)[g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
lim
h→0
g(x + h) + f (x) lim
h→0
g(x + h)− g(x)
h
= f ′(x)g(x) + f (x)g ′(x)
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x)g(x)]′ = lim
h→0
f (x + h)g(x + h)− f (x)g(x)
h
= lim
h→0
f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h)
h
= lim
h→0
[f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)]
h
= lim
h→0
[f (x + h)− f (x)]g(x + h)
h
+ lim
h→0
f (x)[g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
lim
h→0
g(x + h) + f (x) lim
h→0
g(x + h)− g(x)
h
= f ′(x)g(x) + f (x)g ′(x)
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x)g(x)]′ = lim
h→0
f (x + h)g(x + h)− f (x)g(x)
h
= lim
h→0
f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h)
h
= lim
h→0
[f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)]
h
= lim
h→0
[f (x + h)− f (x)]g(x + h)
h
+ lim
h→0
f (x)[g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
lim
h→0
g(x + h) + f (x) lim
h→0
g(x + h)− g(x)
h
= f ′(x)g(x) + f (x)g ′(x)
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x)g(x)]′ = lim
h→0
f (x + h)g(x + h)− f (x)g(x)
h
= lim
h→0
f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h)
h
= lim
h→0
[f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)]
h
= lim
h→0
[f (x + h)− f (x)]g(x + h)
h
+ lim
h→0
f (x)[g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
lim
h→0
g(x + h) + f (x) lim
h→0
g(x + h)− g(x)
h
= f ′(x)g(x) + f (x)g ′(x)
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Aplicando a definic¸a˜o de derivada, temos que
[f (x)g(x)]′ = lim
h→0
f (x + h)g(x + h)− f (x)g(x)
h
= lim
h→0
f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h)
h
= lim
h→0
[f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)]
h
= lim
h→0
[f (x + h)− f (x)]g(x + h)
h
+ lim
h→0
f (x)[g(x + h)− g(x)]
h
= lim
h→0
f (x + h)− f (x)
h
lim
h→0
g(x + h) + f (x) lim
h→0
g(x + h)− g(x)
h
= f ′(x)g(x) + f (x)g ′(x)
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Exemplo 3: Seja a func¸a˜o f (x) = ln x sen x .
f ′(x) = (ln x sen x)′
= (ln x)′ sen x + ln x ( sen x)′
=
1
x
sen x + ln x cos x
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Exemplo 3: Seja a func¸a˜o f (x) = ln x sen x .
f ′(x) = (ln x sen x)′
= (ln x)′ sen x + ln x ( sen x)′
=
1
x
sen x + ln x cos x
Regras Operato´rias das Derivadas
(iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x)
Exemplo 3: Seja a func¸a˜o f (x) = ln x sen x .
f ′(x) = (ln x sen x)′
= (ln x)′ sen x + ln x ( sen x)′
=
1
x
sen x + ln x cos x
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Aplicando a definic¸a˜o de derivada, temos que
[
f (x)
g(x)
]′
= lim
h→0
f (x+h)
g(x+h) − f (x)g(x)
h
= lim
h→0
f(x + h)g(x)− f (x)g(x + h)
hg(x + h)g(x)
= lim
h→0
f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x)
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)
hg(x + h)g(x)
− lim
h→0
f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Aplicando a definic¸a˜o de derivada, temos que
[
f (x)
g(x)
]′
= lim
h→0
f (x+h)
g(x+h) − f (x)g(x)
h
= lim
h→0
f (x + h)g(x)− f (x)g(x + h)
hg(x + h)g(x)
= lim
h→0
f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x)
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)
hg(x + h)g(x)
− lim
h→0
f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Aplicando a definic¸a˜o de derivada, temos que
[
f (x)
g(x)
]′
= lim
h→0
f (x+h)
g(x+h) − f (x)g(x)
h
= lim
h→0
f (x + h)g(x)− f (x)g(x + h)
hg(x + h)g(x)
= lim
h→0
f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x)
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)
hg(x + h)g(x)
− lim
h→0
f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Aplicando a definic¸a˜o de derivada, temos que
[
f (x)
g(x)
]′
= lim
h→0
f (x+h)
g(x+h) − f (x)g(x)
h
= lim
h→0
f (x + h)g(x)− f (x)g(x + h)
hg(x + h)g(x)
= lim
h→0
f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x)
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)
hg(x + h)g(x)
− lim
h→0
f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Aplicando a definic¸a˜o de derivada, temos que
[
f (x)
g(x)
]′
= lim
h→0
f (x+h)
g(x+h) − f (x)g(x)
h
= lim
h→0
f (x + h)g(x)− f (x)g(x + h)
hg(x + h)g(x)
= lim
h→0
f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x)
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
= lim
h→0
[f (x + h)− f (x)]g(x)
hg(x + h)g(x)
− lim
h→0
f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
[
f (x)
g(x)
]′
= lim
h→0
[f (x + h)− f (x)]g(x)
hg(x + h)g(x)
− lim
h→0
f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
=
f ′(x)g(x)
[g(x)]2
− f (x)g
′(x)
[g(x)]2
=
f ′(x)g(x)− f (x)g ′(x)
[g(x)]2
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
[
f (x)
g(x)
]′
= lim
h→0
[f (x + h)− f (x)]g(x)
hg(x + h)g(x)
− lim
h→0
f (x)[g(x + h)− g(x)]
hg(x + h)g(x)
=
f ′(x)g(x)
[g(x)]2
− f (x)g
′(x)
[g(x)]2
=
f ′(x)g(x)− f (x)g ′(x)
[g(x)]2
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Exemplo 4: Seja a func¸a˜o f (x) = tg x .
f ′(x) = ( tg x)′
=
( sen x
cos x
)′
=
( sen x)′ cos x − sen x(cos x)′
[cos x ]2
=
cos x cos x − sen x(− sen x)
cos2 x
=
cos2 x + sen 2x
cos2 x
=
1
cos2 x
= sec2 x
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Exemplo 4: Seja a func¸a˜o f (x) = tg x .
f ′(x) = ( tg x)′
=
( sen x
cos x
)′
=
( sen x)′ cos x − sen x(cos x)′
[cos x ]2
=
cos x cos x − sen x(− sen x)
cos2 x
=
cos2 x + sen 2x
cos2 x
=
1
cos2 x
= sec2 x
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Exemplo 4: Seja a func¸a˜o f (x) = tg x .
f ′(x) = ( tg x)′
=
( sen x
cos x
)′
=
( sen x)′ cos x − sen x(cos x)′
[cos x ]2
=
cos x cos x − sen x(− sen x)
cos2 x
=
cos2 x + sen 2x
cos2 x
=
1
cos2 x
= sec2 x
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Exemplo 4: Seja a func¸a˜o f (x) = tg x .
f ′(x) = ( tg x)′
=
( sen x
cos x
)′
=
( sen x)′ cos x − sen x(cos x)′
[cos x ]2
=
cos x cos x − sen x(− sen x)
cos2 x
=
cos2 x + sen 2x
cos2 x
=
1
cos2 x
= sec2 x
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Exemplo 4: Seja a func¸a˜o f (x) = tg x .
f ′(x) = ( tg x)′
=
( sen x
cos x
)′
=
( sen x)′ cos x − sen x(cos x)′
[cos x ]2
=
cos x cos x − sen x(− sen x)
cos2 x
=
cos2 x + sen 2x
cos2 x
=
1
cos2 x
= sec2 x
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Exemplo 4: Seja a func¸a˜o f (x) = tg x .
f ′(x) = ( tg x)′
=
( sen x
cos x
)′
=
( sen x)′ cos x − sen x(cos x)′
[cos x ]2
=
cos x cos x − sen x(− sen x)
cos2 x
=
cos2 x + sen 2x
cos2 x
=
1
cos2 x
= sec2 x
Regras Operato´rias das Derivadas
(iv)
[
f (x)
g(x)
]′
Exemplo 4: Seja a func¸a˜o f (x) = tg x .
f ′(x) = ( tg x)′
=
( sen x
cos x
)′
=
( sen x)′ cos x − sen x(cos x)′
[cos x ]2
=
cos x cos x − sen x(− sen x)
cos2 x
=
cos2 x + sen 2x
cos2 x
=
1
cos2 x
= sec2 x
Regras Operato´rias das Derivadas
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x)
[f (x)− g(x)]′ = [f (x) + (−1)g(x)]′
= f ′(x) + [(−1)g(x)]′
= f ′(x) + (−1)[g(x)]′
= f ′(x)− g ′(x)
Regras Operato´rias das Derivadas
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x)
[f (x)− g(x)]′ = [f (x) + (−1)g(x)]′
= f ′(x) + [(−1)g(x)]′
= f ′(x) + (−1)[g(x)]′
= f ′(x)− g ′(x)
Regras Operato´rias das Derivadas
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x)
[f (x)− g(x)]′ = [f (x) + (−1)g(x)]′
= f ′(x) + [(−1)g(x)]′
= f ′(x) + (−1)[g(x)]′
= f ′(x)− g ′(x)
Regras Operato´rias das Derivadas
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x)
[f (x)− g(x)]′ = [f (x) + (−1)g(x)]′
= f ′(x) + [(−1)g(x)]′
= f ′(x) + (−1)[g(x)]′
= f ′(x)− g ′(x)
Regras Operato´rias das Derivadas
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x)
Exemplo 5: Seja a func¸a˜o f (x) = x2 −√x .
f ′(x) =
(
x2 −√x)′
=
(
x2
)′ − (√x)′
= 2x − 1
2
√
x
Regras Operato´rias das Derivadas
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x)
Exemplo 5: Seja a func¸a˜o f (x) = x2 −√x .
f ′(x) =
(
x2 −√x)′
=
(
x2
)′ − (√x)′
= 2x − 1
2
√
x
Regras Operato´rias das Derivadas
(v) [f (x)− g(x)]′ = f ′(x)− g ′(x)
Exemplo 5: Seja a func¸a˜o f (x) = x2 −√x .
f ′(x) =
(
x2 −√x)′
=
(
x2
)′ − (√x)′
= 2x − 1
2
√
x