Derivatives of Inverse Functions Worksheet

Derivatives of Inverse Functions Worksheet
  • Page 1
 1.  
If 9z2 - 6zy = 4, then find dzdy using implicit differentiation.
a.
16z
b.
(18z - 6y)6z
c.
(18z + 6y)6z
d.
(18z - 6y)
e.
6z(18z-6y)


Solution:

9z2 - 6zy = 4
[Write the equation.]

ddy(9z2 - 6zy) = ddy(4)
[Differentiate both sides with respect to y.]

18z dzdy - 6(z×1 + y×dzdy) = 0

dzdy =6z(18z - 6y)
[Group the terms and find dzdy.]


Correct answer : (5)
 2.  
If 6a3 - 5y3 = 3ay, then find dyda by using implicit differentiation.
a.
- 18a2 + 3y3a + 15y2
b.
18a2 - 3y3a - 15
c.
18a2 - 3a
d.
3a + 15y218a2 - 3y
e.
18a2 - 3y3a + 15y2


Solution:

6a3 - 5y3 = 3ay
[Write the equation.]

dda(6a3 - 5y3) =dda(3ay)
[Differentiate both sides with respect to a.]

18a2 - 15y2dyda = 3(y + adyda)
[Use the General Power Rule.]

dyda = 18a2 - 3y3a + 15y2
[Group the terms and find dyda.]


Correct answer : (5)
 3.  
If 2e(3c2+4y2) = 3y, then find dydc using implicit differentiation.
a.
6c3 - 8y
b.
12ce(3c2+4y2)3-16ye(3c2+4y2)
c.
e(3c2+4y2)
d.
12c3-16y
e.
- 12c 16y


Solution:

2e(3c2+4y2) = 3y
[Write the equation.]

ddc(2e(3c2+4y2)) =ddc(3y)
[Differentiate on both sides with respect to c.]

2e(3c2+4y2)×(6c+8ydydc) = 3dydc
[Use the Chain Rule.]

12ce(3c2+4y2)+16ye(3c2+4y2)dydc = 3dydc

12ce(3c2+4y2) =dydc(3-16ye(3c2+4y2))
[Group the terms and find dydc.]

dydc = 12ce(3c2+4y2)3 - 16ye(3c2+4y2)


Correct answer : (2)
 4.  
If 89z2 + 7zy + 9y2 = 0, then find dzdy using implicit differentiation.
a.
7 z(178z + 7 y)
b.
- (7 z + 18 y)(178z + 7 y)
c.
7 z + 18 y
d.
(178z + 7 y)(7 z + 18 y)
e.
(7z + 18y)(178z + 7y)


Solution:

89z2 + 7zy + 9y2 = 0
[Write the equation.]

ddy(89z2)+ddy(7zy)+ddy(9y2) = ddy (0)
[Differentiate both sides with respect to y.]

178zdzdy + 7(z + ydzdy) + 18y = 0
[Group the terms.]

dzdy =- (7z + 18y)(178z + 7y)
[Solve for dzdy.]


Correct answer : (2)
 5.  
If 2y cos (3y + 16) = x, then find dydx using implicit differentiation.
a.
- 16y(sin(3y + 16))
b.
12cos(3y+16)-6ysin(3y +16)
c.
12cos(3y+16)+6ysin(3y +16)
d.
12cos (3y+16)
e.
2cos (3y+16) - 6y(sin(3y + 16))


Solution:

2y cos (3y + 16) = x
[Write the equation.]

ddx(2y cos (3y + 16)) = ddx(x)
[Differentiate both sides with respect to x.]

- 6y(sin(3y + 16)dydx + 2cos(3y + 16)dydx = 1
[Use the Product Rule.]

dydx = 12cos (3y+16) - 6ysin(3y + 16)
[Group the terms and find dydx.]


Correct answer : (2)
 6.  
If 9b3 + 5y3 = 6y, then find dydb.
a.
6 - 15y227b2
b.
27b26-y2
c.
276 - 15y2
d.
b26 - 15y2
e.
27b26 - 15y2


Solution:

9b3 + 5y3 = 6y
[Write the equation.]

ddb(9b3+5y3) =ddb(6y)
[Differentiate both sides with respect to b.]

27b2 + 15y2dydb= 6dydb
[Group the terms and find dydb.]

dydb = 27b26 - 15y2


Correct answer : (5)
 7.  
If e2sin (z + 11) + 3zy = 7, then find dzdy using implicit differentiation.
a.
- 3z3y + 2cos (z+11)e2sin (z+11)
b.
 3z3y + 2cos (z+11)e2sin (z+11)
c.
- 3y + 2cos (z+11)e2sin (z+11)3z
d.
Does not exist
e.
- 3ze2sin (z+11)


Solution:

e2sin (z+11)+3zy = 7
[Write the equation.]

ddy (e2sin (z+11)+3zy) =ddy(7)
[Differentiate both sides with respect to y.]

(2cos (z + 11))e2sin (z+11)dzdy + 3z + 3ydzdy = 0
[Use the Product Rule, Chain Rule.]

dzdy = - 3z3y + 2cos (z+11)e2sin (z+11)
[Group the terms and find dzdy.]


Correct answer : (1)
 8.  
If (c + y)2 = 2y - 3c - 5, then find the value of dydc using implicit differentiation.
a.
- 2(c + y) + 3
b.
- 2(c+y) + 32(c+y) - 2
c.
- 2(c+y) - 22(c+y) + 3
d.
12(c+y) - 2
e.
2(c+y) + 32(c+y) - 2


Solution:

(c + y)2 = 2y - 3c - 5
[Write the equation.]

ddc((c + y)2) =ddc(2y - 3c - 5)
[Differentiate both sides with respect to c.]

2(c + y)(1 + dydc) = 2dydc - 3 - 0
[Use the Chain Rule.]

(2(c + y) - 2)dydc = - 3 - 2(c + y)
[Group the terms and find dydc.]

dydc = - (2(c+y) + 3)(2(c+y) - 2)


Correct answer : (2)
 9.  
If cy = cos (6c + 2y), then find dydc using implicit differentiation.
a.
- c + 2sin(6c + 2y)1+6sin(6c + 2y)
b.
- y+6sin(6c + 2y)c + 2sin(6c + 2y)
c.
Does not exist
d.
1+6sin(6c + 2y)c + 2sin(6c + 2y)
e.
2yc + 2sin(6c + 2y)


Solution:

cy = cos (6c + 2y)
[Write the equation.]

ddc(cy) = ddc (cos (6c + 2y))
[Differentiate both sides with respect to c.]

(y + cdydc) = - sin (6c + 2y) ddc (6c + 2y)
[Use the Chain Rule.]

y + cdydc = - sin (6c + 2y)(6 + (2dydc))

dydc(c + 2sin(6c + 2y)) = - (y + 6sin(6c + 2y))
[Group the terms and find dydc.]

dydc = - y+6sin(6c + 2y)c + 2sin(6c + 2y)


Correct answer : (2)
 10.  
If y = 3a3y2 + 9ay3, then find dyda using implicit differentiation.
a.
9y3+9a2y2
b.
11 - 6a3y - 27ay2
c.
9y3+9a2y21 - 6a3y - 27ay2
d.
9y31 - 6a3y
e.
Does not exist


Solution:

y = 3a3y2 + 9ay3
[Write the equation.]

dda (y) = dda (3a3y2) + dda (9ay3)
[Differentiate both sides with respect to a.]

dyda = 3a3(2y)(dyda) + 3y2 (3a2) + 9a(3(y2)(dyda)) + 9y3
[Use the Product Rule.]

dyda (1 - 6a3y - 27ay2) = 9y3 + 9a2y2
[Group the terms and find dyda.]

dyda = 9y3+9a2y21 - 6a3y - 27ay2


Correct answer : (3)

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