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Differentiation Worksheet

Differentiation Worksheet
  • Page 1
 1.  
If f(u) = 5u2-9u9u+4, then f ′(u)
a.
45u2 + 40u - 36(9u + 4)2
b.
10u-99
c.
-41 + 135u2
d.
101u -36(9u+4)2


Solution:

f ′ (u) = (9u + 4)(10u - 9) - (5u2 - 9u)9(9u + 4)2
[Use Quotient rule.]

= 90u2 + 40u - 81u - 36 - 45u2 + 81u(9u + 4)2

= 45u2 + 40u - 36(9u + 4)2


Correct answer : (1)
 2.  
If g(t) = 3t3+4t-54t-5, then g′(t)
a.
24 - 45t2(4t - 5)2
b.
t3 - 45t2(4t - 5)2
c.
24t3 +t2(4t - 5)2
d.
24t3 - 45t2(4t - 5)2


Solution:

g′(t) = (4t - 5)(9t2 + 4) - (3t3 + 4t - 5)(4)(4t - 5)2
[Use the quotient rule and the chain rule.]

= 36t3 - 45t² + 16t - 20 - 12t3 - 16t + 20(4t - 5)2
[Simplify.]

= 24t3 - 45t2(4t - 5)2
[Simplify.]


Correct answer : (4)
 3.  
If y = 6xe2x, then y′ = ?
a.
6xe2x + 6e2x
b.
36x + 12e4x
c.
12x2e2x - 1 + 6e2x
d.
6 e2x (2x + 1)


Solution:

y = 6xe2x

y′ = 6x(2) e2x + e2x (6)
[Use the product rule.]

= 12x e2x + 6e2x

= 6e2x (2x + 1)


Correct answer : (4)
 4.  
If y = - 8 x2e- 2x, then find y′.
a.
16xe-2x(1+4x)
b.
8x e- 2x (2x - 2)
c.
8e-2xx(1 - 2x)
d.
- 8e-8x(x2+2x)


Solution:

y′ = - 8x2.(- 2e- 2x) + e- 2x (- 16x)
[Use product rule.]

= 16x2e- 2x - 16x e- 2x

= 8x e- 2x (2x - 2)


Correct answer : (2)
 5.  
If y = ln(2x)4x2-4, then find y′.
a.
[1-2ln(2x)]-4x(x2 - 4)2
b.
x2[1-2ln(2x])-4x
c.
2ln(2x)]x(4x2 - 4)2
d.
4x2[1-2ln(2x)]-4x(4x2 - 4)2


Solution:

y′ = (4x2 - 4)[22x]-(ln 2x)(8x)(4x2-4)2
[Use the quotient rule.]

= 4x -4x - 8x(ln 2x)(4x2 -4)2

= 4x2[1-2ln(2x)]-4x(4x2 - 4)2
[Multiply numerator and denominator with x and simplify.]


Correct answer : (4)
 6.  
If g(x) = - 5x26x + 5, then what is g′ (x)?
a.
50(6x + 5)2 
b.
x2 + 50(6x + 5)
c.
- 30x2 - 50x(6x + 5)2 
d.
30x(6x + 5)2 


Solution:

g′(x) = (6x + 5)(- 10x) - (- 5x2)(6)(6x + 5)2 
[Use the quotient rule.]

= - 60x2 - 50x + 30x2(6x + 5)2

= - 30x2 - 50x(6x + 5)2 


Correct answer : (3)
 7.  
If y = x-3x + 6, then find y′.
a.
6 - x + 6x2x(x + 6)2
b.
6 + x - 2x2x(x + 6)2
c.
- x + 2x2(x + 2)2
d.
none of the above


Solution:

y′ = (x + 6)[12x-12 ]-(x12 - 3)(x + 6)2
[Use the quotient rule.]

= [12x12 +62x-12 -x12 + 3(x + 6)2] [2 / 2]
[Multiply and divide by 2.]

= x12 + 6x-12 - 2x12 + 62(x + 6)2 x12x12
[ Multiply and divide by x12.]

= - x + 6 + 6x122x12(x + 6)2

= 6 - x + 6x122x12(x + 6)2

= 6 - x + 6x2x(x + 6)2


Correct answer : (1)
 8.  
If y = x2 - 2x + 2x - 2, then y′ = ?
a.
3x2 - 8x + 6
b.
-2(x - 1)(x - 2)2
c.
2x - 2
d.
x2 - 4x + 2(x - 2)2


Solution:

y′ = (x - 2)(2x - 2) - (x2 - 2x + 2)(1)(x - 2)2

= x2- 4x + (4 - 2)(x - 2)2

= x2 - 4x + 2(x - 2)2


Correct answer : (4)
 9.  
If y = 7x2+2x-6x-7, then y′ = ?
a.
x2-98x-8(x-7)2
b.
7x2-98x-8(x-7)2
c.
7x2-8(x-7)2
d.
7x2-98x(x-7)2


Solution:

y′ = (x-7) (14x+2)-(7x2+2x-6)(1)(x-7)2
[Use the quotient rule.]

= 14x2+2x-98x-14-7x2-2x+6(x-7)2

= 7x2-98x-8(x-7)2
[Simplify.]


Correct answer : (2)
 10.  
If y = 9x2-2x-4x + 9, then y′ = ?
a.
18x2-162x+14(x-9)2
b.
9x2 + 162x - 14(x+9)2
c.
18x2(x+9)
d.
162x - 14x+9


Solution:

y′ = (x+9)(18x-2)-(9x2-2x-4)(1)(x+9)2
[Use quotient rule.]

= 18x2+162x-2x-18-9x2+2x+4(x+9)2

= 9x2+162x-14(x+9)2
[Simplify.]


Correct answer : (2)

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