[CALCULO 1 - UNIVATES] - Técnicas de diferenciação - calcular derivadas
Autor: Dizzy | Publicado: 2011-06-10T10:09:00-03:00

Centro
 Universitário UNIVATES
Cursos:&nbsp;Engenharias e Ciências Exatas 
Cálculo&nbsp;I – Semestre A/2011
Professora&nbsp;Dra.Ana Cecília Togni


TÉCNICAS DE DIFERENCIAÇÃO
&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;Alguns teoremas importantes que permitem que se calcule a derivada de forma mais simples são os seguintes:

T1) A derivada de uma função constante é
0, isto é, se c for um número real qualquer então: f’(c) = 0.
T2) (Regra da Potência). Se n for um
número inteiro positivo e f( x) = xn,
então
&nbsp;f’(x) = n xn-1
T3) Se f for diferenciavel
em x e c for um numero real qualquer então cf é
também diferenciável em x e se expressa assim : f’(x)= cf’
T4) (Derivadas das somas e Diferenças). Se
f e g forem diferenciáveis em x então f+g e f-g também o são ou seja :
[f(x) + g(x) ]’ =f ’(x) + g’(x)
[f(x) – g(x)]’ = f’(x) – g’(x)
T5) (Regra do Produto) Se f e g forem
diferenciáveis em x então o produto f. g também o é. Ou seja:
[f(x). g(x) ] =f(x).g’(x) + g(x).f’(x)
T6) (Regra do Quociente) Se f e g forem
diferenciáveis em x e g’(x) for diferente de zero, então f/g é diferenciável em x e:
[f(x)/g(x) =g(x).f’(x)– f(x) g’(x)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;[g(x)]2

T7)(Regra do Recíproco) Se a função g for diferenciavel em x e g(x) diferente de zero então 1/g é diferenciavel em x e [1/g(x)]’ = g’(x)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[g(x)]2&nbsp;
T8) Se n for qualquer inteiro, então :&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;[xn]’ = nxn-1

T9) (Regra da Cadeia) Se g for diferenciavel no ponto x e f for diferenciavel&nbsp; no ponto g(x), então a composição f e g
também é diferenciavel no ponto x.&nbsp; Além disso se y = f(g(x)) e u =g(x) então y =
f(u) e 
dy = dy . du&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ou&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;f’(u) = y’(u).u’(x)dx&nbsp;&nbsp;&nbsp;&nbsp; du&nbsp; &nbsp;dx
T10)Derivadas de Funções Trigonométricas
i)Se f(x)= sen x&nbsp;&nbsp; então f’(x) = cosx
ii)Se f(x) = cos x então
f’(x) = - sen x 
iii) Se f(x) = tgx x então
f’(x) = sec2 x
iV) Se f(x) = sec x então
f’(x)= secx .tgx
v)Sef(x) = cotg x então f’(x) = - cossec2 x 
vi)Se f(x) = cossec x então
f’(x) = -cossecx.cotgx
vii)Se f(x) = arc senx então f’(x ) =&nbsp;1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;√1– x2

viii) Se f(x) = arc cos x então f’(x) = -&nbsp;&nbsp;1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; √1 –x2&nbsp; 
ix)Se f(x) = arc tgx então f’(x) = &nbsp;1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;1 + x2
x) Se f(x) = arc cotg então f’(x) = - 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;1 + x2
xi)Se f(x) = arc séc| x | então f’(x) = 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; x√x2 -1
xii)Se f(x) = arc cossec |x| então f’(x) =&nbsp;&nbsp;-&nbsp; 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;x√x2-1



Observação:As duas últimas formulas não se referem a
funções elementares, mas&nbsp; cada uma delas
resume duas funções elementares para x &gt;0 e x&lt;0


T11)Derivadas das Funções logarítmicas
T12)Se f(x)= ln |
x| então f’(x) =&nbsp; 1&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;para&nbsp;x ≠ 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;x
No caso de f(x) ser função composta fica assim: f(x)
= ln u&nbsp; então
f’(x ) = 1 . du&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; u&nbsp; &nbsp;dx&nbsp;&nbsp;
T13) Se f(x) =loga
x&nbsp;&nbsp;&nbsp; então f’(x) =&nbsp; 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; x ln a
No caso de f(x) ser composta fica assim: f(x) = loga u = 1 .&nbsp;&nbsp;&nbsp;&nbsp; du&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;u ln a&nbsp;&nbsp;&nbsp; dx



T14)Derivada de funções exponenciais
i) Se f(x) = ax
então f’(x) = ax.lna
No caso de função composta fica assim: f(x) = au então f’(x) = au.lna.du&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp;dx

&nbsp;&nbsp; ii) Se f(x)= ex então f’(x) = ex
&nbsp;Quando for
função composta fica assim: f(x) = eu então f’(x) = eu.du&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp;dx