TABLA DE DERIVADAS E INTEGRAIS DERIVADAS f(x)=g(x)+h(x) f(x)=k.g(x) f’(x)=g’(x)+h’(x) f’(x)=k.g’(x) f(x)=g(x).h(x) f’(x)=g’(x).h(x)+g(x).h’(x) TIPO f ( x) = INTEGRAIS g ( x) h( x ) f '( x) = f ( x ) = g[h( x )] SIMPLE g '( x ). h( x) − g ( x ). h'( x ) ∫ kf ( x)dx = k ∫ f ( x)dx ∫ [ f ( x) + g( x)]dx = ∫ f ( x)dx + ∫ g( x)dx [h ( x ) ] 2 f '( x ) = g ' [h( x )]. h'( x) COMPOSTA SIMPLE COMPOSTA CONSTANTE f(x)=k f’(x)=0 ∫ kdx = kx + C IDENTIDADE f(x)=x f’(x)=1 ∫ dx = x + C POTENCIAL EXPONENCIAL TANXENTE f '( x ) = e x f ( x) = e g ( x ) f '( x ) = e g ( x ) . g '( x ) ∫ e dx = e f ( x) = a x f '( x ) = a x .ln a f ( x) = a g ( x ) f '( x ) = a g ( x ) .ln a. g '( x ) ∫ a x dx = f ( x ) = ln x f ' ( x) = n −1 n 1 x f ( x ) = ln g ( x ) 1 x.ln a f ( x ) = log a g ( x ) 1 g '( x ) . g '( x ) = g( x) g( x) 1 g '( x ) f '( x ) = . g '( x ) = g ( x ).ln a g ( x ).ln a f '( x ) = f '( x ) = cos x f ( x ) = seng ( x ) f '( x ) = cos g ( x ). g '( x ) f ( x ) = cos x f '( x ) = − senx f ( x ) = cos g ( x ) f ' ( x ) = − seng ( x ). g ' ( x ) f ( x) = tgx f '( x) = 1 = 1 + tg 2 x cos2 x f ( x ) = arc tgx f '( x ) = ARCO SENO f ( x ) = arc senx f '( x ) = f ( x ) = arc cos f '( x ) = 1 1+ x2 1 1− x2 −1 1− x2 f ( x ) = tg[ g ( x )] f ( x ) = arc tg[ g ( x )] f '( x ) = f '( x ) = f ( x ) = arc sen[ g ( x )] f '( x ) = f ( x ) = arc cos[ g ( x )] Si n ≠ − 1 f '( x ) = 1 g '( x ) . g '( x ) = cos g ( x ) cos2 g ( x ) 2 1 1 + [ g ( x )] 2 1 1 − [ g ( x )] 2 −1 1 − [ g ( x )] 2 . g '( x ) = . g' ( x) = . g' ( x) = x n dx = x f ( x ) = senx ARCO TANXENTE ARCO COSENO ∫ f ( x) = e x f ( x ) = log a x f ' ( x ) = COSENO f '( x ) = n.[ g ( x )] . g '( x ) f '( x ) = n. x n −1 LOGARÍTMICA SENO f ( x ) = [ g ( x )] f ( x) = x n 1 ∫ cos 2 1 + [ g ( x )] 1 − [ g ( x )] − g' ( x) 1 − [ g ( x )] 2 ax +C ln a ∫ senxdx = − cos x + C ∫ cos xdx = senx + C ∫ ∫ x dx = n n+ 1 ∫e +C f ( x) ∫ +C . f '( x )dx = e f ( x ) + C ∫ a f ( x ) . f ' ( x)dx = 1 2 2 ∫ ∫ x dx = ln x + C g' ( x) g '( x ) x f ( x) [ f ( x)] . f '( x)dx = [ ] n +1 x n+1 +C n+ 1 a f ( x) +C ln a f '( x ) dx = ln f ( x ) + C f ( x) ∫ senf ( x). f '( x)dx = − cos f ( x) + C ∫ cos f ( x). f '( x)dx = senf ( x) + C 2 ∫ (1 + tg x)dx = tgx + C ∫ cos2 f ( x) dx = ∫ (1 + tg [ f ( x)]). f '( x)dx = tg[ f ( x)] + C 2 1 ∫ 1 + x 2 dx = arc tgx + C 1 dx = arc senx + C 1− x2 −1 dx = arc cos x + C 1− x2 f '( x) f ' ( x) ∫ 1 + [ f ( x)] dx = arc tgf ( x) + C 2 ∫ ∫ f ' ( x) 1 − [ f ( x)] 2 − f '( x ) 1 − [ f ( x )] 2 dx = arc senf ( x ) + C dx = arc cos f ( x ) + C