Universidade Tecnológica Federal do Paraná - UTFPR

Universidade Tecnológica Federal do Paraná
UTFPR — Campus Pato Branco
Exercícios de Derivadas de Funções Reais de Variável Real
1. Usando a definição de derivadas f ′ f(x + ∆x) − f(x)
(x) = lim
ou f ′ f(x) − f(p)
(p) = lim
,
∆x→0 ∆x
x→p x − p
calcule a derivada das seguintes funções nos pontos dados:
(a) f(x) = 2x 2 − 3x + 4, p = 2
(b) f(x) = 3 x , p = 1
2
(c) f(t) = 3√ t, p = 8
(d) g(x) = cos x, p = π 2
(e) f(x) = 3 sin x, p = 2π e 0
(f) v = √ 3 − 2 √ t; t = 4
t
(g) f(x) = 5x − x 2 , f ′ (−3) e f ′ (0)
(h) f(x) = x + 9 x , x = −3
2. Calcule a derivada das funções abaixo usando as propriedades adequadas:
(a) f(x) = 16x 3 − 4x 2 + 3
(b) f(x) = −5x 3 + 21x 2 − 3x + 4
(c) f(x) = 5
(d) f(t) = 2t − 1
(e) y = 8
(f) y = 2x + 1
(g) y = 5√ x 2 − 4√ x 3 + x 4
(h) y = x 4 5 − x 1 6
(i) f(x) = 10 1001000
(j) s(t) = 5t − 1
2t − 7
(k) f(x) = 3 x + 2√ x − 1
4 √ x
(l) f(r) = 4 r 2 + 5 r 3
(m) f(x) = (2x 2 − 1) · (1 − 2x)
(n) y = (x 2 − 3x 4 ) · (x 5 − 1)
(o) g(t) = 5t − 2
1 + t + t 2
(p) f(x) = (x 2 + 3x + 3) · (x + 3)
(q) f(x) =
2x3
4x + 2
(r) y = (x + 2) · (x 5 − 6x)
(s) g(x) = x2 − 4
x + 0, 5
(t) r = 2 · ( 1 √
θ
+ √ θ)
(u) f(x) =
1
(x 2 − 1) · (x 2 + x + 1)
(v) v = (1 − t) · (1 − t 2 ) −1
(w) y = √ x + 1
3√
x
4
3. Calcule a derivada das funções trigonométricas abaixo usando as regras de derivação:
(a) f(x) = tan x = sin x
cos x
(b) g(x) = sec x = 1
cos x
(c) f(x) = √ x · (2 sin x + x 2 )
(e)
(d) h(θ) = π sin θ − cos θ
2
y = x 3 − 1 2 cos x
(f) y = 5
(2x) 3 + 2 sin x
(g) y = 3 x + 5 sin x
(h) y =
cotg x
1 + cotg x
4. Calcule a derivada das funções exponenciais e logarítmicas abaixo usando as regras de
derivação:
1

(a) f(x) =
ex
cos x
(b) y = e x · sin x
(c) f(x) = x 2 · ln x
(d) f(x) = (x 2 + 1) · e x
(e) y =
ex
2e x + 1
(f) y = xe x − e x
(g) y = x 2 e x − xe x
(h) y = 2e x
5. Usando a regra do quociente e do produto, ache dy
dx
(a) y = 2x − 1
(c) y =
x + 3
(b) y = 4x + 1
x 2 − 5
(i) y = e −t (t 2 − 2t + 2)
no ponto x = 1:
( ) 3x + 2
· (x −5 + 1)
x
( ) x + 1
x − 1
(d) y = (2x 8 − x 678 ) ·
6. Resolva e determine se é verdadeiro ou falso, se g(x) = x 5 , então lim
x→2
g(x) − g(2)
x − 2
7. Resolva e determine se é verdadeiro ou falso:
= 80.
(a)
(b)
d
dx (10x ) = x10 x−1
d
dx (ln 10) = 1 10
(c)
(d)
d
dx (tan2 x) = d
dx (sec2 x)
d
dx |x2 + x| = |2x + 1|
8. Derive utlizando as regras de derivação.
(a) y = sin 4x
(b) y = cos 5x
(c) y = e 3x
√ x + 1(2 − x)
5
(d) y =
(x + 3) 7
(e) y = sin t 3
(f) g(t) = ln(2t + 1)
(g) x = e sin t
(h) f(x) = cos(e x )
(i) y = (sin x + cos x) 3
(j) y = √ (3x + 1)
√ (x ) − 1
(k) y = 3 x + 1
(l) y = tan 2 (sin θ)
(m) x = ln(t 2 + 3t + 9)
(n) f(x) = e tan x
(o) y = sin(cos x)
(p) g(t) = (t 2 + 3) 4
(q) f(x) = cos(x 2 + 3)
(r) y = √ (x + e x )
(s) y = √ t · ln(t 4 )
(t) y = sin(tan √ 1 + x 3 )
(u) y = x · e 3x
(v) y = e x · cos 2x
(w) y = e −x · sin x
(x) y = e 2t · sin 3t
(y) f(x) = e −x2 + ln(2x + 1)
(z) g(t) = et − e −t
e t + e −t
cos 5x
(a 1 ) y =
sin 2x
(b 1 ) f(x) = (e −x + e x2 ) 3
(c 1 ) y = t 3 · e −3t
(d 1 ) y = (sin 3x + cos 2x) 3
(e 1 ) y = √ x 2 + e −x
(f 1 ) y = x · ln(2x + 1)
(g 1 ) y = [ln(x 2 + 1)] 3
(h 1 ) y = ln(sec x + tan x)
(i 1 ) f(x) = ln(x 2 + 8x + 1)
(j 1 ) f(x) = √ 6x + 2
(k 1 ) f(x) = x 4 · e 3x
(l 1 ) f(x) = sin 4 x
2

(m 1 ) f(x) = 5 tan 2x
(e 2 ) y = 2x √ x 2 + 1
(n 1 ) f(x) = (2x 3 − 3x) · (5 − x 2 ) 3
(f 2 ) y = ex
3
1 + x 2
(o 1 ) f(x) = −√ 3x − 5
sin 2θ
(g 2 ) y = e
(p 1 ) y = e x2 +x+1
(h 2 ) y = e mx cos nx
(q 1 ) y = sin 2x · cos x
(i 2 ) y = √ x cos √ x
(r 1 ) y = (2x 2 − 4x + 1) 8
(s 1 ) q = √ (x 2 + 1) 4
(j 2 ) y =
2r − r 2
(2x + 1) 3 (3x − 1) 5
( ) ( )
3πt 3πt
1
(t 1 ) s = sin + cos
(k 2 ) y =
2
2
sin(x − sin(x))
(u 1 ) h(x) = x tan(2 √ x) + 7
(l 2 ) y = ln(cossec 5x)
(v 1 ) r = sin(θ 2 ) cos(2θ)
sec 2θ
(m 2 ) y =
(w 1 ) y = (4x + 3) 4 (x + 1) −3
1 + tan 2θ
(n
(x 1 ) y = x tan −1 2 ) y = e cx (c sin x − cos x)
(4x)
(o
(y 1 ) y = e cos x + cos(e x 2 ) y = ln(x 2 e x )
)
(z 1 ) y = cotg (3x 2 (p
+ 5)
2 ) y = sec(1 + x 2 )
(a 2 ) y = ex
(q 2 ) y = (1 − x −1 ) −1
e −x + 1
1
(r 2 ) y = √ √
(b 2 ) y = (x 4 − 3x 2 + 5) 3
3
(x + x)
(c 2 ) y = cos(tan x)
(s 2 ) y = √ sin √ x
(d 2 ) y = √ 3x − 2
(t 2x + 1 2 ) y = ln(sin x) − 1 2 sin2 x
9. Derive utilizando a derivada implícita:
(a) xy 4 + x 2 y = x + 3y
(b) x 2 cos y + sin 2y = xy
(c) sin(xy) = x 2 − y
(d) y = xe y − y − 1
(e) y 2 + x 2 = 1
(f) y 3 + yx 2 = sen x + 3y 2 x
10. Encontre a derivada das seguintes funções:
(a) y = 8 x
cossec (x)
(b) y = 3
(c) y = x (x2 +1)
(d) y = 7 x2 +2x
(e) y = 3 x ln x
(f) y = log 5 (1 + 2x)
(g) y = (cos x) x
(h) y = x sinh x 2
(i) y = ln(cosh 3x)
(j) y = cosh −1 (sinh x)
tan πθ
(k) y = 10
(l) y = x · tanh −1 √ x
11. Derive utilizando a derivada inversa:
(a) y = (arcsin 2x) 2
(b) y = arctan(arcsin √ x)
(c) y = √ x
(d) y = ln x
3

Respostas
1. (a) f ′ (2) = 5
(b) f ′ (1) = −6
(c) f ′ (8) = 1
(
12
π
)
(d) g ′ = −1
2
2. (a) f ′ (x) = 48x 2 − 8x
(b) f ′ (x) = −15x 2 + 42x − 3
(c) f ′ (x) = 0
(d) f ′ (x) = 2
(e) y ′ = 0
(f) y ′ = 2
(g) y ′ = 2
5 5√ x 3 − 3
4 4√ x + 4x3
(h) f ′ (x) = 4
5 5√ x − 1
6 6√ x 5
(i) f ′ (x) = 0
(j) s ′ (t) =
−33
(2t − 7) 2
(k) f ′ (x) = − 3 x + √ 1 + 1
2 x 8x √ x
(l) f ′ −8r − 15
(r) =
r 4
(m) f ′ (x) = 2(−6x 2 + 2x + 1)
(e) f ′ (2π) = 3
(f) v ′ (4) = − 11
16
(g) f ′ (−3) = 11 e f ′ (0) = 5
(h) f ′ (−3) = 0
(n) y ′ = x(−27x 7 + 7x 5 + 12x 2 − 2)
(o) g ′ (t) = 7 − 5t2 + 4t
(1 + t + t 2 ) 2
(p) f ′ (x) = x 2 + 4x + 4
(q) f ′ (x) = x2 (4x + 3)
(2x + 1) 2
(r) y ′ = 3x 5 + 5x 4 − 6x − 6
(s) g ′ (x) = x2 + x + 4
(x + 0, 5) 2
(t) r ′ = 1 √
θ
− 1
θ √ θ
(u) f ′ (x) =
(v) v ′ = t2 − 2t − 1
(1 + t 2 ) 2
−4x 3 − 3x 2 + 1
((x 2 − 1)(x 2 + x + 1)) 2
(w) y ′ = 1
2 √ x − 4
3x 2 3√ x
3. (a) f ′ (x) = sec 2 x
(b) g ′ (t) = tan t · sec t
(c) f ′ (x) =
2 sin x + x2
2 √ x
(e)
(d) h ′ (θ) = π cos θ + sin θ
2
+ 2 √ x(cos x + x)
y ′ = 3x 2 + 1 2 sin x
(f) y ′ = − 15
8x 4 + 2 cos x
(g) y ′ = 5 cos x − 3 x 2
(h) y ′ 1
= −
2 sin x cos x + 1
4. (a) f ′ (x) = ex (sin x + cos x)
(f) y ′ = e x · x
cos 2 x
(b) f ′ (x) = e x (sin x + cos x)
(g) y ′ = e x (x 2 + x − 1)
(c) f ′ (x) = x(2 ln x + 1)
(d) f ′ (x) = e x (x 2 (h) y ′ = 2e x
+ 2x + 1)
(e) y ′ e x
=
(2e x + 1) 2 (i) y ′ = (−t2 + 4t − 4)
e t
5. (a) y ′ (1) = 7 16
(b) y ′ (1) = − 13
8
6. Verdadeira
4
(c) y ′ (1) = −29
(d) Descontínua em x = 1

7.
5

(a) Falsa
(b) Falsa
8. (a) y ′ = 4 cos 4x
(b) y ′ = −5 sin 5x
(c) y ′ = 3e 3x
(b) y ′ =
(e) y ′ = 3t 2 cos t 3
(f) g ′ (t) = 2
2t + 1
(g)
(h)
1(2 − x) 5
2 √ x + 1(x + 3) 7 − 5√ x + 1(2 − x) 4
(x + 3) 7 − 7√ x + 1(2 − x) 5
(x + 3) 8
x ′ = e sin t cos t
f ′ (x) = −e x sin e x
(i) y ′ = 3(sin x + cos x) 2 (cos x − sin x)
3
(j) y ′ =
2 √ 3x + 1
(k)
√ ( )
y ′ 2
x + 1 2
=
3(x + 1) 2 · 3
x − 1
(l)
y ′ = 2 tan(sin θ) sec 2 (sin θ) cos θ
(m) x ′ 2t + 3
=
t 2 + 3t + 9
(n)
f ′ (x) = e tan x sec 2 x
(o) y ′ = − sin x cos(cos x)
(p) g ′ (t) = 8t(t 2 + 3) 3
(q) f ′ (x) = −2x sin(x 2 + 3)
(r) y ′ = 1 + ex
2 √ x + e x
(s) y ′ = (ln(t4 ) + 4)
2 √ t ln(t 4 )
(t) y ′ = 3x2 cos(tan √ 1 + x 3 ) sec 2 √ 1 + x 3
(u) y ′ = e 3x (1 + 3x)
2 √ 1 + x 3
(v) y ′ = e x (cos 2x − 2 sin 2x)
(w) y ′ = e −x (cos x − sin x)
(x) y ′ = e −2t (3 cos 3t − 2 sin 3t)
(y) f ′ 2
(x) =
2x + 1 − 2xe−x2
(z) g ′ 4e 2t
(t) =
(e 2 t + 1) 2
(a 1 ) y ′ −5 sin 5x sin 2x − 2 cos 5x cos 2x
=
sin 2 2x
(b 1 ) f ′ (x) = 3(e −x + e x2 ) 2 (−e −x + 2xe x2 )
(c 1 ) y ′ = 3t 2 e −3t (1 − t)
(d 1 ) y ′ = 3(sin 3x + cos 2x) 2 (3 cos 3x − 2 sin 2x)
(e 1 ) y ′ 2x − e−x
=
2 √ x 2 + e −x
(f 1 ) y ′ 2x
= ln(2x + 1) +
(2x + 1)
(g 1 ) y ′ = 6x[ln(x2 + 1)] 2
(h 1 ) y ′ = sec x
x 2 + 1
(i 1 ) y ′ 2x + 8
=
x 2 + 8x + 1
(j 1 ) f ′ (x) =
3
√ 6x + 2
(k 1 ) f ′ (x) = e 3x x 3 (4 + 3x)
9. (a) y ′ = 1 − y4 − 2xy
4xy 3 + x 2 − 3
(c) Verdadeira
(d) Falsa
(l 1 ) f ′ (x) = 4 sin 3 x cos x
(m 1 ) f ′ (x) = 10 sec 2 2x
(n 1 ) f ′ (x) = (5 − x 2 ) 2 [(6x 2 − 3)(5 − x 2 ) − 6x(2x 3 − 3x)]
(o 1 ) f ′ 9
(x) =
2 √ (3x − 5) 3
(p 1 ) y ′ = e x2 +x+1 (2x + 1)
(q 1 ) y ′ = 2 cos 2x cos x − sin 2x sin x
(r 1 ) y ′ = 32(2x 2 − 4x + 1) 7 (x − 1)
(s 1 ) q ′ = 1 − r
√
2r − r 2
(t 1 ) s ′ = 3π 2 cos ( 3πx
2
)
− 3π 2
( ) 3πx
sin
2
(u 1 ) h ′ (x) = tan(2 √ x) + √ x sec 2 (2 √ x)
(v 1 ) r ′ = 2θ cos θ 2 cos 2θ − 2 sin θ 2 sin 2θ
(w 1 ) y ′ = (4x + 3)3 (4x + 7)
(x + 1) 4
(x 1 ) y ′ = e x sin e x − e cos x sin x
(y 1 ) y ′ = 5 sec 5x
(z 1 ) y ′ = −6x · cossec 2 (3x + 5)
(a 2 ) y ′ = (2e2x + e 3x )
(e x + 1) 2
(b 2 ) y ′ = 6x(x 4 − 3x 2 + 5) 2 (2x 2 − 3)
(c 2 ) y ′ = − sin(tan x) sec 2 x
(d 2 ) y ′ =
3x + 5
√ 2x + 1(2x + 1)
(e 2 ) y ′ = 2(2x2 + 1)
√
x 2 + 1
(f 2 ) y ′ = ex (1 + x 2 − 2x)
(1 + x 2 ) 2
(g 2 ) y ′ = 2e sin 2θ cos 2θ
(h 2 ) y ′ = e mx (m cos nx − n sin nx)
(i 2 ) y ′ = 1
2 √ x (cos √ x − √ x sin √ x)
(j 2 ) y ′ = cotan 4x − 4x · cossec 4x
(k 2 ) y ′ cos x − cos(x − sin x)
=
sin 2 (x − sin x)
(l 2 ) y ′ = −5cotan 5x
(m 2 ) y ′ 2 sec 2θ(tan 2θ − 1)
=
(1 + tan2θ) 2
(n 2 ) y ′ = e cx (c 2 sin x + sin x)
(o 2 ) y ′ = 2 + x
x
(p 2 ) y ′ = 2x sec(1 + x 2 ) tan(1 + x 2 )
(q 2 ) y ′ 1
= −
(x − 1) 2
(r 2 ) y ′ = − 1 2 √ x + 1
√
6
√ x
3
(x + √ x) 4
(s 2 ) y ′ = cos√ x
4 √ x sin √ x
(t 2 ) y ′ = (cotan x − sin x cos x) = cos3 x
(u 2 ) y ′ = − (x2 + 1) 3 (x 2 + 56x + 9)
(2x + 1) 4 (3x − 1) 6
(c) y ′ =
(2x − y cos xy)
x cos xy + 1
sin x
(b) y ′ =
y − 2x cos y
2 cos 2y − x 2 sin y − x
(d) y ′ =
ey
2 − xe y
6

(e) y ′ = −x
y
10. (a) y ′ = 8 x ln(8)
(b) y ′ = −3 cossec x ln(3)cossec x · cotan x
(c) y ′ = (x + 2 + 1)x x2 + x x2 +1 ln x · 2x
(d) y ′ = 7 x2 +2x ln(7)(2x + 2)
(e) y ′ = 3 x ln x ln 3(ln x + 1)
(f) y ′ 2
=
(1 + 2x) ln 5
(g) y ′ = cos x x(ln(cos x) − x tan x)
11. (a) y ′ = 4 arcsin(2x) √
1 − 4x
2
(b) y ′ =
cos √ x
2 √ x(1 + sin 2 √ x)
(f) y ′ = cos x + 3y2 − 2xy
3y 2 + x 2 − 6y
(h) y ′ = sinh(x 2 ) + 2x 2 cosh(x 2 )
(i) y ′ 3 sinh 3x
=
cosh 3x
(j) y ′ sinh(sinh(x)) cosh x
= −
cosh(sinh(x)) 2
(k) y ′ = π10 tan πx sec 2 πx ln 10
(l) y ′ = 2 tanh √ x − √ xsech 2√ x
2 tanh 2 √ x
(c) y ′ = 1
2 √ x
(d) y ′ = 1 x
Coletânea de exercícios elaborada pelos professores:
• Dra. Dayse Batistus;
• Msc. Ana Munaretto;
• Msc. Cristiane Pendeza;
• Msc. Adriano Delfino;
• Msc. Marieli Musial Tumelero
Digitação:
• 1 a versão: Acadêmico Bruno Brito.
• Versão atual: Acadêmica Larissa Hagedorn Vieira.
Referência Bibliográfica:
ANTON, H., BIVENS, I. e DAVIS, S. Cálculo. vol. 1. Tradução: Claus I. Doering. 8 ed.
Porto Alegre: Bookman, 2007.
GUIDORIZZI, H. L. Um curso de cálculo, vol.1 e 2. 5 a ed. LTC Editora, Rio de Janeiro,
RJ: 2002.
LEITHOLD, L. O cálculo com geometria analítica. Vol.1. 3 a ed. São Paulo: Harbra,
1994.
LIMA, J. D. Apostila de Cálculo I. UTFPR, Pato Branco, 2008.
STEWART, James. Cálculo. Vol. 2. 6 a ed. São Paulo: Pioneira Thomson Learning, 2009.
SWOKOWSKI, E. W. Cálculo com geometria analítica. Vol. 1. 2 a ed. São Paulo: Makron
Books do Brasil,1994.
THOMAS, G. B. Cálculo. Vol. 1. 10 a ed. São Paulo: Person, 2002.
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