REGRAS DE DERIVAÇÃO

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REGRAS DE DERIVAÇÃO 
 
PROPRIEDADES DA DERIVAÇÃO: C = constante 
 
[C . f(x)]’ = C . f[C . f(x)]’ = C . f[C . f(x)]’ = C . f[C . f(x)]’ = C . f ’(x)’(x)’(x)’(x) 
 
[ f(x) [ f(x) [ f(x) [ f(x) ± g(x)]’ = fg(x)]’ = fg(x)]’ = fg(x)]’ = f ’(x) ’(x) ’(x) ’(x) ± gggg ’(x)’(x)’(x)’(x) 
 
[ f(x) . g(x)]’ = ( u . v )’ = u’.v + u.v’[ f(x) . g(x)]’ = ( u . v )’ = u’.v + u.v’[ f(x) . g(x)]’ = ( u . v )’ = u’.v + u.v’[ f(x) . g(x)]’ = ( u . v )’ = u’.v + u.v’ 
 
[ f(x) / g(x)]’ = ( u / v )’ = (u[ f(x) / g(x)]’ = ( u / v )’ = (u[ f(x) / g(x)]’ = ( u / v )’ = (u[ f(x) / g(x)]’ = ( u / v )’ = (u ’.v ’.v ’.v ’.v ---- u.vu.vu.vu.v ’’’’ ) / v²) / v²) / v²) / v² 
 
REGRA DA CADEIA :REGRA DA CADEIA :REGRA DA CADEIA :REGRA DA CADEIA : [ f(g(x))]’ =[ f(g(x))]’ =[ f(g(x))]’ =[ f(g(x))]’ = [ f(u) ]’ = f[ f(u) ]’ = f[ f(u) ]’ = f[ f(u) ]’ = f ’(u) . u’’(u) . u’’(u) . u’’(u) . u’ 
 
 
DERIVADA PADRÃO APLICANDO A REGRA DA CADEIA 
 
nnnn = constante 
 
�� =
��
��
=
�(� )
��
= (� )� = � !" 
��
��
=
��
�#
∙
�#
��
= (# )� = # !". #′ 
 
 
(&' )� = & ' (&#)� = &#. #′ 
 
(() ' )′ = 
"
'
 (() # )′ = 
"
#
∙ #′ 
 
( *& ' )� = +,- ' (*& # )� = +,- # . #′ 
 
 
(+,- ' )′ = −*& ' (+,- # )′ = −*& # . #′ 
 
Exemplos: 
 
/) (0�1)� = 0. 1. �1!" = 2�0 
 
3) (4�)� = (4�")� = 4. ". �"!" = 4. ". �5 = 4. ". " = 4 
 
6) (1)� = (1. �5)� = 1. 5. �5!" = 1. 5. �!" = 5 
 
Regra da cadeia 
 
�) [(0�7 − �4)1]′ → # = (0�7 − �4) → (#1)′ = # !". #′ = 1(0�7 − �4)1!". (0�7 − �4)′ 
 
= 1(0�7 − �4)0. ("4�2 − 4�1) 
 
&) 9&!7'
0
:
�
 → # = −7'0 → (&#)� = &#. #� = &!7'
0
. (−7'0)� = &!7'
0
. (−"4')