# Derivatives of Inverse Functions Worksheet

Derivatives of Inverse Functions Worksheet
• Page 1
1.
If 9$z$2 - 6$\mathrm{zy}$ = 4, then find $\frac{dz}{dy}$ using implicit differentiation.
 a. $\frac{1}{6z}$ b. c. d. (18z - 6$y$) e. $\frac{6z}{\left(18z-6y\right)}$

#### 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.]

2.
If 6$a$3 - 5$y$3 = 3$a$$y$, then find $\frac{dy}{da}$ by using implicit differentiation.
 a. b. c. 18 d. e.

#### 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.]

3.
If 2${e}^{\left(3{c}^{2}+4{y}^{2}\right)}$ = 3$y$, then find $\frac{dy}{dc}$ using implicit differentiation.
 a. b. $\frac{12c{e}^{\left(3{c}^{2}+4{y}^{2}\right)}}{3-16y{e}^{\left(3{c}^{2}+4{y}^{2}\right)}}$ c. ${e}^{\left(3{c}^{2}+4{y}^{2}\right)}$ d. $\frac{12c}{3-16y}$ e. - $\frac{12c}{16y}$

#### 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)

4.
If 89$z$2 + 7$\mathrm{zy}$ + 9$y$2 = 0, then find $\frac{dz}{dy}$ using implicit differentiation.
 a. b. c. 7 $z$ + 18 $y$ d. e.

#### 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.]

5.
If 2$y$ cos (3$y$ + 16) = $x$, then find $\frac{dy}{dx}$ using implicit differentiation.
 a. - b. c. d. e. 2cos (3$y$+16) - 6$y$(sin(3$y$ + 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.]

6.
If 9$b$3 + 5$y$3 = 6$y$, then find $\frac{dy}{db}$.
 a. b. $\frac{27{b}^{2}}{6-{y}^{2}}$ c. d. e.

#### 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

7.
If = 7, then find $\frac{dz}{dy}$ using implicit differentiation.
 a. - b. c. - d. Does not exist e.

#### 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.]

8.
If ($c$ + $y$)2 = 2$y$ - 3$c$ - 5, then find the value of $\frac{dy}{dc}$ using implicit differentiation.
 a. - 2($c$ + $y$) + 3 b. - c. - d. e.

#### 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)

9.
If $\mathrm{cy}$ = cos (6$c$ + 2$y$), then find $\frac{dy}{dc}$ using implicit differentiation.
 a. - b. - c. Does not exist d. e.

#### 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)

10.
If $y$ = 3$a$3$y$2 + 9$a$$y$3, then find $\frac{dy}{da}$ using implicit differentiation.
 a. ${9y}^{3}+9{a}^{2}{y}^{2}$ b. c. d. 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