CEGEP CHAMPLAIN - ST. LAWRENCE
201-103-RE: Differential Calculus
Patrice Camiré
Problem Sheet #26
Higher Derivatives
1. Simplify/rewrite y and then find the corresponding higher derivative.
(Hint: Use long division for (c) and (d).)
6x3 − 5x + 1
x2
5
if y =
2x − 3
3x2 − 11x − 12
if y =
x+1
x5 − 2x3 + x + 3
x−1
√
if y = ln(x x + 3)
(a) Find y (5) if y =
(d) Find y (5) if y =
(b) Find y (3)
(e) Find y (6)
(c) Find y (6)
√
4
(f) Find y (4) if y = ln x3 2x + 1 ex
2. Complete the following statements about the graph of the function y = f (x).
(a) If y 0 =
dy
> 0, then the graph of y = f (x) is ...
dx
(b) If y 0 =
dy
< 0, then the graph of y = f (x) is ...
dx
(c) If y 0 =
dy
= 0, then the graph of y = f (x) is ...
dx
(d) If y 00 =
d 2y
> 0, then the graph of y = f (x) is ...
dx2
(e) If y 00 =
d 2y
< 0, then the graph of y = f (x) is ...
dx2
3. In each case, find all values of x for which the first derivative is zero or undefined. Moreover, find
all values of x for which the second derivative is zero or undefined.
(a) y =
x2
(x − 1)2
(b) y =
x2
x−1
4. (a) Find y (1) if y = x.
(x − 3)2
(x + 1)2
(x + 2)3
(d) y =
(x − 1)2
(c) y =
(e) y =
x2
(x − 3)4
(f) y = xe−2x
(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+ .
Answers
1. (a) y = 6x − 5x−1 + x−2 and y (5) =
(b) y = 5(2x − 3)−1 and y (3) = −
120(5x − 6)
x7
240
(2x − 3)4
(c) y = 3x − 14 + 2(x + 1)−1 and y (6) =
1440
(x + 1)7
360
(d) y = x4 + x3 − x2 − x + 3(x − 1)−1 and y (5) = −
(x − 1)6
1
2
1
(6)
(e) y = ln(x) + ln(x + 3) and y = −60
+
2
x6 (x + 3)6
1
3
8
4
(4)
(f) y = 3 ln(x) + ln(2x + 1) + x and y = −6
+
−4
2
x4 (2x + 1)4
2. (a) increasing.
(b) decreasing.
(c) flat.
−2x
, x = 0, 1
(x − 1)3
2(2x + 1)
, x = −1/2, 1
y 00 =
(x − 1)4
x(x − 2)
(b) y 0 =
, x = 0, 2, 1
(x − 1)2
2
y 00 =
, x=1
(x − 1)3
8(x − 3)
(c) y 0 =
, x = 3, −1
(x + 1)3
−16(x − 5)
y 00 =
, x = 5, −1
(x + 1)4
3. (a) y 0 =
4. (a) 1
(d) concave up.
(e) concave down.
(x + 2)2 (x − 7)
, x = −2, 7, 1
(x − 1)3
54(x + 2)
y 00 =
, x = −2, 1
(x − 1)4
(d) y 0 =
−2x(x + 3)
, x = 0, −3, 3
(x − 3)5
√
6(x2 + 6x + 3)
y 00 =
, x = −3 ± 6, 3
6
(x − 3)
(e) y 0 =
(f) y 0 = −e−2x (2x − 1) , x = 1/2
y 00 = 4e−2x (x − 1) , x = 1
(d) 1 · 2 · 3 · 4
(b) 1 · 2
(e) 1 · 2 · 3 · 4 · 5
(c) 1 · 2 · 3
(f) 1 · 2 · · · (n − 1) · (n) = n!
We call n! n factorial.
Additional Problems
5. (a) Find y (5) if y = xπ .
(i) Find y (3) if y = log3 (2x − 1).
(b) Find y (2) if y = cos(x3 ).
p
(c) Find y (3) if y = x2 + 1.
√
(d) Find y (3) if y = x.
(j) Find y (3) if y = 2−x .
(k) Find y (3) if y = cos2 (x).
(l) Find y (4) if y = x2 −
(e) Find y (3) if y = sin(2x).
(f) Find y (2) if y = tan(3x).
(m) Find y (3) if y =
(g) Find y (3) if y = arctan(−x).
2
1
.
x
4x3 − x2 + 3x − 1
.
x
(n) Find y (2) if y = (3x2 − x + 1)5 .
(h) Find y (2) if y = ex .
Answers
5. (a) y (5) = π(π − 1)(π − 2)(π − 3)(π − 4)xπ−5
(b) −3x[2 sin(x3 ) + 3x3 cos(x3 )]
(c)
(d)
(x2
−3x
+ 1)5/2
3
8x5/2
2
(h) 2ex (2x2 + 1)
(i)
16
ln(3)(2x − 1)3
(j) − ln3 (2)2−x
(k) y (3) = 8 sin(x) cos(x)
(e) −8 cos(2x)
(l) y (4) = −
(f) 18 tan(3x) sec2 (3x)
(g)
3x2 )
2(1 −
(x2 + 1)3
(m) y (3) =
24
x5
6
x4
(n) y (2) = 10(3x2 − x + 1)3 (81x2 − 27x + 5)