CEGEP CHAMPLAIN - ST. LAWRENCE
201-NYA-05: Differential Calculus
Patrice Camiré
Higher Derivatives
1. (a) Find y (5) if y =
(b) Find y (3) if y =
6x3 − 5x + 1
x2
(c) Find y (6) if y =
3x2 − 11x − 12
x+1
5
2x − 3
(d) Find y (5) if y =
x5 − 2x3 + x + 3
x−1
√
2. (a) Find y (6) if y = ln(x x + 3)
√
4
(b) Find y (4) if y = ln x3 2x + 1 ex
3. (a) Find y (2) if y = x sin(x).
(b) Find y (3) if y = cos2 (x).
(c) Find y (4) if y = x2 −
1
.
x
(d) Find y (3) if y = sin(x2 ).
4. (a) Find y (1) if y = x.
(c) Find y (5) if y = ln
xesin(2x)
√
3
x−1
!
(d) Find y (3) if y = ln(sin3 (2x) cos2 (3x))
p
π
x2 + 1 − .
3
π
if y = x .
(e) Find y (2) if y =
(f) Find y (5)
4x3 − x2 + 3x − 1
.
x
if y = (3x2 − x + 1)5 .
(g) Find y (3) if y =
(h) Find y (2)
(d) Find y (4) if y = x4 .
(b) Find y (2) if y = x2 .
(e) Find y (5) if y = x5 .
(c) Find y (3) if y = x3 .
(f) Find y (n) if y = xn , where n ∈ Z+ .
5. Find y (2) using implicit differentiation.
(a) x3 + y 3 = 1
(f) x1/3 − y 1/4 = 5
(b) x4 + y 4 = 3
(g) x2 + sin(y) = 3
1
√
(c) 3x + 2 y =
2
(h) x + tan(y) = −1
(d) sin(x) + sin(y) = −1
(e) tan(x/y) = π
(i) x2 + y = 1 − sin(y)
√
(j) cot(y) + sin(x) = 2
6. Complete the following statements about the graph of the function y = f (x).
(a) If
dy
> 0, then the graph of y = f (x) is ...
dx
(b) If
dy
< 0, then the graph of y = f (x) is ...
dx
(c) If
dy
= 0, then the graph of y = f (x) is ...
dx
(d) If
d 2y
> 0, then the graph of y = f (x) is ...
dx2
(e) If
d 2y
< 0, then the graph of y = f (x) is ...
dx2
7. If y = sin(x), find y (n) (0) for any n ≥ 0.
8. If y = cos(x), find y (n) (0) for any n ≥ 0.
9. (Additional Problems)
(a) Find y (4) if y = x5 .
(h) Find y (3) if y = sin(2x).
(b) Find y (2) if y = cos(x3 ).
(i) Find y (2) if y = tan(3x).
x2
.
x+1
p
if y = x2 + 1.
√
if y = ln(x2 3 3x + 4).
(c) Find y (5) if y =
(d) Find y (3)
(e) Find y (4)
(f) Find y (2) if y = 2x6 − 3x4 + x2 − 5.
√
(g) Find y (3) if y = x.
(j) Find y (3) if y = arctan(−x).
2
(k) Find y (2) if y = ex .
(l) Find y (3) if y = log3 (2x − 1).
(m) Find y (3) if y = 2−x .
(n) Find y (2) if y = x sin(x).
Answers
1. (a) y = 6x − 5x−1 + x−2 and y (5) =
(b) y = 5(2x − 3)−1 and y (3) = −
(c) y = 3x − 14 +
120(5x − 6)
x7
240
(2x − 3)4
2
1440
and y (6) =
x+1
(x + 1)7
(d) y = x4 + x3 − x2 − x +
3
360
and y (5) = −
x−1
(x − 1)6
2
1
+
x6 (x + 3)6
3
8
1
4
(4)
+
−4
(b) y = 3 ln(x) + ln(2x + 1) + x and y = −6
2
x4 (2x + 1)4
1
3
1
(c) y = ln(x) + sin(2x) − ln(x − 1) and y (5) = 8
+
4
cos(2x)
−
3
x5
(x − 1)5
1
2. (a) y = ln(x) + ln(x + 3) and y (6) = −60
2
(d) y = 3 ln(sin(2x)) + 2 ln(cos(3x)) and y (3) = 12 4 csc2 (2x) cot(2x) − 9 sec2 (3x) tan(3x)
3. (a) y (2) = 2 cos(x) − x sin(x)
(b) y (3) = 8 sin(x) cos(x)
(c) y (4) = −
24
x5
(d) y (3) = −12x sin(x2 ) − 8x3 cos(x2 )
4. (a) 1
(b) 1 · 2
(c) 1 · 2 · 3
(d) 1 · 2 · 3 · 4
(e) 1 · 2 · 3 · 4 · 5
(f) 1 · 2 · · · (n − 1) · (n) = n!
We call n! n factorial.
(e) y (2) = (x2 + 1)−3/2
(f) y (5) = π(π − 1)(π − 2)(π − 3)(π − 4)xπ−5
(g) y (3) =
6
x4
(h) y (2) = 10(3x2 − x + 1)3 (81x2 − 27x + 5)
5. (a) −
(b) −
(c)
2x 2x4
− 5
y2
y
(f)
√
4 y
8y 3/4
−
3x4/3 9x5/3
3x2 3x6
− 7
y3
y
(g)
4x2 tan(y) − 2 cos(y)
cos2 (y)
9
2
(h) −2 sin(y) cos3 (y)
sin(x) sin(y) cos2 (x)
(d)
+
cos(y)
cos3 (y)
(e) 0
−2
4x2 sin(y)
+
1 + cos(y) (1 + cos(y))3
h
i
(j) sin2 (y) 2 cos2 (x) cos(y) sin(y) − sin(x)
(i)
6. (a) increasing.
(b) decreasing.
(c) flat.
(d) concave up.
(e) concave down.
7. y (n) (0) =
8. y
(n)


0 , if n = 2k , k ∈ N
1 , if n = 1 + 4k , k ∈ N

−1 , if n = 3 + 4k , k ∈ N


0 , if n = 2k + 1 , k ∈ N
1 , if n = 4k , k ∈ N
(0) =

−1 , if n = 2 + 4k , k ∈ N
9. (a) 120x
(b) −3x[2 sin(x3 ) + 3x3 cos(x3 )]
(c)
(d)
−120
(x + 1)6
(x2
(e) −
−3x
+ 1)5/2
12
162
−
4
x
(3x + 4)4
(f) 60x4 − 36x2 + 2
(g)
3
8x5/2
(h) −8 cos(2x)
(i) 18 tan(3x) sec2 (3x)
(j)
2(1 − 3x2 )
(x2 + 1)3
2
(k) 2ex (2x2 + 1)
(l)
16
ln(3)(2x − 1)3
(m) − ln3 (2)2−x
(n) 2 cos(x) − x sin(x)