# 1/4 Derivadas Outras funções: 1) Função exponencial: y = ex

Propaganda
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Outras fun&ccedil;&otilde;es:
y
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

1) Fun&ccedil;&atilde;o exponencial: y = ex

x
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Gr&aacute;fico da fun&ccedil;&atilde;o y = ex
y
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
x
x
(e )’ = e
x
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y
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2) Fun&ccedil;&atilde;o logar&iacute;tmica: y = ln x
x
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Gr&aacute;fico da fun&ccedil;&atilde;o y = ln x
y
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x
1
(ln x)’ =
x
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3) Fun&ccedil;&otilde;es trigonom&eacute;tricas:
y



a) Fun&ccedil;&atilde;o seno: y = sen x

x
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Gr&aacute;fico da fun&ccedil;&atilde;o y = sen x
y
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x
(sen x)’ = cos x
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y
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b) Fun&ccedil;&atilde;o cosseno: y = cos x
x
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Gr&aacute;fico da fun&ccedil;&atilde;o y = cos x
y
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(cos x)’ = - sen x
x
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c) Fun&ccedil;&atilde;o tangente: y = tg x =
(tg x)’ =
senx
cos x
cos x.( senx )' senx.(cos x)' cos x. cos x  senx.( senx)
=
cos 2 x
(cos x) 2
1
cos 2 x  sen 2 x
 1 
2
(tg x)’ =
=
=
 = sec x
2
2
cos
x
cos
x
cos x


2
d) Fun&ccedil;&atilde;o cotangente: y = cotg x =
cos x
senx
(cotg x)’ =
senx.(cos x)' cos x.( senx)' senx.( senx)  cos x.(cos x)'
=
sen 2 x
( senx) 2
(cotg x)’ =
1
1
 sen 2 x  cos 2 x  ( sen 2 x  cos 2 x)
=
=
=
2
2
2
sen x
sen 2 x
sen x
sen x
1 
(cotg x)’ =  

 senx 
2
=  cos sec 2 x
e) Fun&ccedil;&atilde;o secante: y = sec x =
(sec x)’ =
1
cos x
1 senx
cos x.1'1.(cos x)' 0  1.( senx) 1.senx
.
=
=
=
2
2
2
cos x cos x
(cos x)
(cos x)
(cos x)
(sec x)’ = sec x.tgx
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f) Fun&ccedil;&atilde;o cossecante: y = cossec x =
(cossec x)’ =
1
senx
 1 cos x
senx.1'1.( senx )' 0  1. cos x
.
=
=
2
2
senx senx
( senx )
( senx )
(cossec x)’ =  cos sec x. cot gx
Exerc&iacute;cios
1
x
a) f ( x)  ln x  e x → f ' ( x)   e x
1
x
b) f ( x)  5 ln x  2e x → f ' ( x)  5.  2e x
c) f ( x)  6senx  3tgx → f ' ( x)  6 cos x  3 sec 2 x
d) f ( x) 
1
cot gx
1
cos sec 2 x
cos x 
→ f ' ( X )   senx 
3
2
3
2
e) f ( x)  10 sec x 
cos sec x
cos sec x. cot gx
→ f ' ( x)  10 sec x.tgx 
4
4
f) f ( x)  e x .tgx → f ' ( x)  e x . sec 2 x
g) f ( x)  5 ln x  2 cos sec x → f ' ( x) 
5
 2 cos sec x. cot gx
x
h) f ( x) 
ln x
1
→ f ' ( x) 
x
e
x.e x
i) f ( x) 
cos x  2senx
senx  2 cos x
→ f ' ( x) 
sec 2 x
tgx
j) f ( x)  senx.5 cos sec x → f ' ( x)  5 cos x. cos sec x. cot gx
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