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CEGEP CHAMPLAIN - ST. LAWRENCE
201-103-RE: Differential Calculus
Patrice Camiré
Problem Sheet #13
Rules of Differentiation
1. Find the derivative of the following functions.
(a) 3x2 − 5x + 2
(d) 3x10 − x9 − x + 1
√
(e) x2 − x + 3
(b) 6x3 + x2 − 8x + 9
6
3
2
5
(c) −x + 7x − 4x − 3
2
(f) x + 4x − 2
(g) 2x2 + 3x + π 2
(h) x100 − x50 + 6x2 −
3
7
√
11
2
(i) 5x − 3x + x − 98
2. Find the derivative of the following functions. Do not use the product rule or the quotient rule.
√
√ √
√
√
4x2 + x + 3
(e) x x − 2 3 x 4 x + 7
(a) x−3 + 2 x − x2
(h)
x2
1
3
8
1
3
(b)
+
+√ +
x
1
4
x x2
x 2
(f) √ − 2 + 8
x2 − 8x + 2
x
x
x
√
(i)
x3 x−2
1
x
(c) 5 − 2 + √
2
3
x
x
x
2x − 5x − 1
(g)
2x3 + x − 9
√
√
√
4
3
5
x
(j)
(d) 12 x − 6 x + 20 x
x4
3. (i) Multiply the expression and find the derivative.
(ii) Find the derivative using the product rule.
(d) (x4 + 1)(x3 + 1)
(a) (2x + 3)(4x − 1)
(g) x(5x + 1)(2x + 3)
2
(b) (5x − 1)(3x − 2)
(e) (3x − x + 2)(4x + 1)
2
2
(c) (x + 1)(2x − 1)
3
2
(f) (x − x + 3)(x + x + 1)
(h) (2x − 1)(3x + 1)(4x + 1)
(i) (x2 + 1)(x + 1)(x − 2)
4. Find the derivative of the following functions.
x
2x + 1
3x − 1
(b)
4x + 5
(a)
(c)
x2 + 1
x+1
(e)
x2 − x + 2
x2 + 2x + 1
(g)
5x2 − x + 1
2x + 3
(d)
x3 − 2
4x − 1
(f)
x3 − 1
x3 + 1
(h)
x3 − x + 1
x2 + 1
5. Find the derivative of the following functions.
(a) (x2 + 3x + 1)4
(b) (x4 − 2x − 3)3
(c) (x2 + 5)4 + 9
8
(d) 1 − (3x + 1)10
3
p
(2x + 1)4 + x
p
(f) 3 (4x2 + 1)3 − x6
(e)
6. Find the derivative of the following functions.
(a) x(3x2 − 4x + 1)4
(c) (x3 + 2x − 1)2 (x3 + 2x + 1)2
(b) (x2 + 1)3 (x2 − 5)4
(d) (5x2 + 3x − 1)3 (x2 − 4x + 8)10
7. (i) Find the derivative of the function.
(ii) Find all values of x for which the tangent line to the function is horizontal.
(3x + 1)2
(a)
x2 + 1
(4x − 1)5
(d)
(x + 1)3
(b)
(2x − 1)3
(6x + 1)2
(e)
(5x + 1)6
(2x + 1)4
(c)
(x2 + 2)4
(x2 + 3)8
(f)
(x2 − 1)3
(x2 − 9)4
3x + 2 8
(g)
4x + 1
s
x2 + 1
(h)
x2 − 1
6x − 1 4
(i)
5x − 2
(j)
x2 − 3x + 1
x2 − 1
10
Answers
(d) 30x9 − 9x8 − 1
(g) 4x + 3
(b) 18x + 2x − 8
(e) 2x − 1
(h) 100x99 − 50x49 + 12x
(c) −6x5 + 21x2 − 8x
(f) 5x4 + 8x
(i) 35x6 − 6x + 1
1. (a) 6x − 5
2
2. (a) −
1
3
+ √ − 2x
4
x
x
(b) −
6
1
−
− 4x−3/2
x2 x3
(c) −
2
4
1
+
− x−4/3
x3 x5 3
(d) 3x−3/4 − 2x−2/3 + 4x−4/5
3√
7
(e)
x − x−5/12
2
6
5 3/2
8
8
(f) x + 3 − 9
2
x
x
1
(g) 2 + 2
x
(h) −
(i)
6
1
− 3
2
x
x
3√
4
x − √ − x−3/2
2
x
(j) −
2
3
36
−
+
x2 x4 x5
(d) x2 (7x4 + 4x + 3)
(g) 30x2 + 34x + 3
(b) 30x − 13
(e) 36x2 − 2x + 7
(h) 72x2 + 4x − 5
(c) 6x2 − 2x + 2
(f) 5x4 + 6x2 + 8x − 1
(i) 4x3 − 3x2 − 2x − 1
3. (a) 16x + 10
4. (a)
1
(2x + 1)2
(c)
x2 + 2x − 1
(x + 1)2
(e)
3x2 − 2x − 5
(x2 + 2x + 1)2
(g)
5(2x2 + 6x − 1)
(2x + 3)2
(b)
19
(4x + 5)2
(d)
8x3 − 3x2 + 8
(4x − 1)2
(f)
6x2
(x3 + 1)2
(h)
x4 + 4x2 − 2x − 1
(x2 + 1)2
(e)
8(2x + 1)3 + 1
p
2 (2x + 1)4 + x
5. (a) 4(2x + 3)(x2 + 3x + 1)3
(b) 6(2x3 − 1)(x4 − 2x − 3)2
(c) 64x(x2 + 5)3 (x2 + 5)4 + 9
7
(d) −90(3x + 1)9 1 − (3x + 1)10
2
(f)
2x 4(4x2 + 1)2 − x4
((4x2 + 1)3 − x6 )2/3
6. (a) (3x2 − 4x + 1)3 (27x2 − 20x + 1)
(b) 2x(x2 + 1)2 (x2 − 5)3 (7x2 − 11)
(c) 4x(x2 + 2)(3x2 + 2)(x3 + 2x − 1)(x3 + 2x + 1)
(d) (5x2 + 3x − 1)2 (x2 − 4x + 8)9 (130x3 − 251x2 + 64x + 112)
7. (a) (i)
2(3x + 1)(3 − x)
(x2 + 1)2
(ii) x = 3, −
1
3
6(2x + 3)(2x − 1)2
(b) (i)
(6x + 1)3
1 3
(ii) x = , −
2 2
−8x(x2 + 1)(x2 + 2)3
(c) (i)
(x2 + 3)9
(ii) x = 0
(4x − 1)4 (8x + 23)
(d) (i)
(x + 1)4
1 23
(ii) x = , −
4
8
(e) (i)
2(10x + 11)(5x + 1)5
(2x + 1)5
1 11
(ii) x = − , −
5 10
(f) (i)
−2x(x2 + 23)(x2 − 1)2
(x2 − 9)5
(ii) x = 0, 1, −1
(g) (i)
−40(3x + 2)7
(4x + 1)9
(ii) x = −
(h) (i) √
x2
2
3
−2x
+ 1(x2 − 1)3/2
(ii) x = 0
(i) (i)
−28(6x − 1)3
(5x − 2)5
(ii) x =
1
6
10(x2 − 3x + 1)9 (3x2 − 4x + 3)
(x2 − 1)11
√
3± 5
(ii) x =
2
(j) (i)