Baixe Lista de Exercícios Resolvidos: Derivadas em Cálculo a Uma Variável e outras Exercícios em PDF para Cálculo Diferencial e Integral, somente na Docsity!
IBM0718 - Cálculo a Uma Variável Ibmec
3 ª LISTA DE EXERCÍCIOS: Derivadas
0.1 Regras de Derivação
- Obtenha as funções derivadas das funções f ( x ) a seguir:
(a) f ( x ) = 3
(b) f ( x ) = −
(c) f ( x ) = π + 4
(d) f ( x ) = x (e) f ( x ) = x^7
(f) f ( x ) =
x^8
(g) f ( x ) =
x
(h) f ( x ) =
x (i) f ( x ) = 5
x
(j) f ( x ) =
√ (^7) x
(k) f ( x ) = 4
x^3 (l) f ( x ) = xπ
(Resp: (a) f ′( x ) = 0; (b) f ′( x ) = 0; (c) f ′( x ) = 0; (d) f ′( x ) = 1; (e) f ′( x ) = 7 x^6 ; (f) f ′( x ) = − 8 x −^9 = − (^) x^89 ; (g) f ′( x ) = 12 x −^12 = 2 √^1 x ; (h) f ′( x ) = − 12 x −^32 = − 2 √^1 x 3 ; (i) f ′( x ) = 15 x −^45 = 5 √ (^51) x 4 ; (j) f ′( x ) = − 17 x −^87 = − 7 √ (^71) x 8 ; (k) f ′( x ) = 34 x −^14 = 4 √^34 x ; (l) f ′( x ) = πxπ −^1 )
- Obtenha as funções derivadas das funções f ( x ) a seguir:
(a) f ( x ) = 2 x
(b) f ( x ) =
) x
(c) f ( x ) = ex (d) f ( x ) = log( x )
(e) f ( x ) = log 2 ( x ) (f) f ( x ) = log 13 ( x ) (g) f ( x ) = ln( x ) (h) f ( x ) = log 0 , 25 ( x )
(Resp: (a) f ′( x ) = 2 x^ ln(2); (b) f ′( x ) =
( 1 5
) x ln
( 1 5
) ; (c) f ′( x ) = ex ; (d) f ′( x ) = (^) x ln(10)^1 ; (e) f ′( x ) = (^) x ln(2)^1 ; (f) f ′( x ) = (^) x ln^1 ( 13 ) ; (g) f ′( x ) = (^1) x ; (h) f ′( x ) = (^) x ln(0^1 , 25) )
- Obtenha as funções derivadas das funções a seguir:
(a) f ( x ) = x^3 + sen( x ) − 3 (b) g ( u ) = ln( u ) − arccos( u ) (c) h ( y ) = log( y ) + cotg( y ) − π
(d) r ( q ) =
q
(e) S ( t ) = arctg( t ) − sec( t ) +
(f) g ( x ) = log 35 ( x ) + πx^ − 1 (g) η ( p ) =
p^3 − 3
p^8 + 2 e (h) α ( θ ) = cossec( θ ) − tg( θ ) + 4
(Resp: (a) f ′( x ) = 3 x^2 +cos( x ); (b) g ′( u ) = (^) u^1 + √ 1 −^1 u 2 ; (c) h ′( y ) = (^) y ln(10)^1 − cossec^2 ( y ); (d) r ′( q ) = − (^) q^12 + eq^ ; (e) S ′( t ) = (^) 1+^1 t 2 − sec( t ) tg( t ); (f) g ′( x ) = (^) x ln^1 ( 35 ) + πx^ ln( π ); (g) η ′( p ) = 3
√ p 2 −^
8 √^3 p^5 3 ; (h) α ′( θ ) = − cossec( θ ) cotg( θ ) − sec^2 ( θ ))
- Obtenha as funções derivadas das funções a seguir:
(a) f ( x ) = 3 cos( x )
(b) g ( t ) =
ln( t ) 5
(c) η ( y ) =
y^6
(d) u ( h ) = 6 · 3 h^ −
eh 6
(e) h ( p ) =
p^3 3
p
πp 7
(f) T ( t ) =
√ (^3) t 2
4
− π · tg( t )
(Resp: (a) f ′( x ) = −3 sen( x ); (b) g ′( t ) = (^51) t ; (c) η ′( y ) = − 24 y −^7 = − (^24) y 7 ; (d) u ′( h ) = 6 · 3 h^ ln(3) − eh 6 ; (e)^ h ′( p ) =^ p^2 −^
1 4 √ p^3 −^
πp^ ln( π ) 7 ; (f)^ T^ ′( t ) =^
1 6 √^3 t −^ π^ sec
(^2) ( t ))
- Utilizando a regra do produto , obtenha as funções derivadas das funções a seguir:
(a) y ( t ) = t^3 · et (b) p ( q ) = ln( q ) · eq (c) y ( t ) = t −^3 · sen( t ) (d) g ( t ) = cos( t ) · sen( t ) (e) g ( x ) = x · log 6 ( x ) (f) S ( t ) = 3 t^2 · arctg( t ) (g) T ( x ) = tg( x ) · 3 x
(h) y ( u ) = ( u^3 + 2 u ) · eu
(i) f ( x ) = ln( x ) · (2 x − x^4 )
(j) y ( x ) = [ln( x ) + ex ] · sen( x )
(k) y ( t ) = ( t^2 + 2 t − 2) ·
( 2 t^3 −
t
)
(l) β ( θ ) =
[√
θ + tg( θ ) − 3
] · πθ
(Resp: (a) y ′( t ) = 3 t^2 · et^ + t^3 · et ; (b) p ′( q ) = (^1) q · eq^ + ln( q ) · eq^ = e qq + ln( q ) · eq^ ; (c) y ′( t ) = − 3 t −^4 · sen( t ) + t −^3 · cos( t ) = − (^) t (^4) · sen(^3 t ) + t −^3 · cos( t ); (d) g ′( t ) = − sen^2 ( t ) + cos^2 ( t ); (e) g ′( x ) = log 6 ( x ) + (^) ln(6)^1 ; (f) S ′( t ) = 6 t · arctg( t ) + 3 t^2 · (^) 1+^1 t 2 = 6 t · arctg( t ) + (^) 1+^3 t^2 t 2 ; (g) T ′( x ) = sec^2 ( x ) · 3 x^ + tg( x )· 3 x^ ln(3); (h) y ′( u ) = (3 u^2 +2)· eu^ +( u^3 +2 u )· eu ; (i) f ′( x ) = (^1) x ·(2 x − x^4 )+ln( x )·(2− 4 x^3 ); (j) y ′( x ) =
( (^1) x +^ ex
) · sen( x )+[ln( x ) + ex ]·cos( x ); (k) y ′( t ) = (2 t +2)·
( 2 t^3 − (^5) t
) +( t^2 +2 t −2)·
( 6 t^2 + (^) t^52
) ; (l) β ′( θ ) =
[ (^1) 2 √ θ + sec
(^2) ( θ )
] · πθ^ +
[√ θ + tg( θ ) − 3
] · πθ^ ln( π ))
- Utilizando a regra do quociente (ou da divisão ), obtenha as funções derivadas das funções a seguir:
(a) y ( x ) =
2 x − 1 x^2 + 5
(b) y ( t ) =
1 − t^3 t^2 + t (c) y ( x ) =
x^3 + 2 (d) f ( x ) =
cos( x ) ex
(e) g ( u ) =
tg( u ) 3 u + 1
(f) g ( ω ) =
ln( ω ) ω
(g) T ( α ) =
1 + cos( α ) sen( α )
(Resp: (a) y ′( x ) = 2( x
(^2) +5)−(2 x −1)2 x ( x^2 +5)^2 =^ −^2 x
(^2) +2 x + ( x^2 +5)^2 ; (b)^ y ′( t ) =^
− 3 t^2 ( t^2 + t )−(1− t^3 )(2 t +1) ( t^2 + t )^2 =^ − t
(^4) − 2 t (^3) − 2 t − 1 ( t^2 + t )^2 ; (c) y ′( x ) = (^) ( x − (^36) +2) x^22 ; (d) f ′( x ) = −^ sen( x ) e
x −cos( x ) ex e^2 x^ =^ −^
[ sen( x )+cos( x )] ex^ ; (e)^ g ′( u ) =^
sec^2 ( u )(3 u +1)−3 tg( u ) (3 u +1)^2 ; (f) g ′( ω ) = 1 − ω ln( 2 ω ); (g) T ′( α ) = −^1 sen−cos( (^2) ( αα ) ))
- Utilizando a regra da cadeia , obtenha as funções derivadas das funções a seguir:
(a) y ( x ) = sen(3 x ) (b) g ( u ) = ln( u^3 − 2 u ) (c) y ( x ) = e sen( x ) (d) f ( t ) = tg (sec( t )) (e) η ( x ) = arcsen( x^3 − 1) (f) F ( u ) = arctg ( sen( u ))
(g) T ( x ) = 6
5 x (h) R ( s ) = −20 log 7 ( s^2 − s + 2) (i) f ( t ) = sen^5 ( t ) (j) L ( p ) =
√ ln( p ) (k) p ( q ) = ln( eq^ + √^3 q ) (l) β ( θ ) = cotg
( 5 θ^ − ln( θ ) + θ^3
)
- Para cada função f ( x ) a seguir, determine a equação da reta tangente ao gráfico no ponto P indicado.
(Resp: (a) y = 4 x + 1; (b) y = 3 x − π ; (c) y = 2 x − 5 ; (d) y = − 4 x − 5 )
