[implicit differentiation][section 11]

(1.)   x3 + y3 = 3             here is the problem

(3x2) + (3y2)(dy/dx) = 0      take the derivative implicitly

(x2) + (y2)(dy/dx) = 0   divide thru by 3, cancel

-x2              -x2     subtract x2 from each side
_________________________
(y2)(dy/dx) = -x2     subtract
___________   ______
y2          y2       divide each side by y2

dy/dx = -x2/y2            cancel

(2.)   x3 + y3 = xy             here is the problem

3x2 + (3y2)(dy/dx) = x(dy/dx) + y

[take the derivative implicitly]

-x(dy/dx)   -x(dy/dx)  subt this fr ea side
____________________________________
3x2 + (3y2)(dy/dx) - x(dy/dx) = y          subtract

-3x2                               - 3x2   subt fr ea side
___________________________________________
(3y2)(dy/dx) - x(dy/dx) = y - 3x2      subtract

(dy/dx)(3y2 - x) = y - 3x2               factor
__________________  ________
3y2 - x        3y2 - x          divide ea side by this

(dy/dx) = (y - 3x2)/(3y2 - x)           cancel
_    _
(3.)   √x + √y = 2              here is the problem

x1/2 + y1/2 = 2      use 1/2 power  for radical signs

(1/2)x-1/2 + (1/2)y-1/2(dy/dx) = 0

[take the derivative implicitly]

x1/2 + y-1/2 (dy/dx) = 0       multiply thru by 2, cancel

-x1/2                - x1/2    subtract x1/2 from each side
______________________________
y-1/2  (dy/dx) = -x1/2         subtract

dy/dx = -(xy)1/2     multiply each side by y1/2, cancel

(4.)  (1/x) + (1/y) = 1          here is the problem

-x-2 - y-2(dy/dx) = 0      take the derivative implicitly

y2 + x2(dy/dx) = 0    multiply thru by -(xy)2, cancel

-y2            -y2   subtract y2 from each side
_________________________
x2(dy/dx) = -y2         subtract
_________   ___
x2       x2        divide each side by x2

dy/dx = -(y/x)2            divide and cancel

(5.)  x-7/8 + y-7/8 = 7/8           here is the problem

(-7/8)x-15/8 - (7/8)y-15/8
* (dy/dx) = 0

[take the derivative implicitly]

x-15/8 + y-15/8 * (dy/dx) = 0   multiply thru by -8/7, cancel

y15/8 + x15/8 (dy/dx) = 0    multiply thru by (xy)15/8, cancel

-y15/8                -y15/8     subtract this fr ea side
______________________________
x15/8 (dy/dx) = -y15/8         subtract

dy/dx = -(y/x)15/8    divide each side by x15/8, cancel

(6.)   x3/4 + y3/4 = 2             here is the problem

(3/4)x-1/4 + (3/4)y-1/4 * (dy/dx) = 0

[take the derivative
implicitly]

x-1/4 + y-1/4
* (dy/dx) = 0    multiply thru by 4/3, cancel

y1/4 + x1/4 * (dy/dx) = 0    multiply thru by x1/4 * y1/4

-y1/4                  - y1/4   subt this fr ea side
___________________________________
x1/4 * (dy/dx) = -y1/4       subtract

dy/dx = -(y/x)1/4   div ea side by x1/4, cancel

(7.)  (3xy + 1)5 = x2            here is the problem

5(3xy + 1)4 * [3x(dy/dx) + 3y] = 2x

[take the derivative implicitly][use the chain rule]

____________________________  __________________
5(3xy + 1)4                 5(3xy + 1)4

[divide each side by this]

3x(dy/dx) + 3y = (2x)/[5(3xy + 1)4]  cancel

-3y               - 3y  subt this fr ea
_______________________________________________ side

2x
3x(dy/dx) =  _________________ - 3y    subtract
5(3xy + 1)4

2                   divide each side
dy/dx =   ____________________ - (y/x)   by 3x, cancel
15(3xy + 1)4

__
(8.)   x2 - √xy + y2 = 6             here is the problem

x2 - x1/2y1/2 + y2 = 6     use the 1/2 power for radical

2x - (1/2)x-1/2y1/2 - (1/2)x1/2y-1/2 * (dy/dx)  + 2y(dy/dx) = 0

[take the derivative implicitly]

_____________________________________________this to ea side

2x - (1/2)(x/y)1/2 * (dy/dx) + 2y(dy/dx) = (1/2)(y/x)1/2  add

4x - (x/y)1/2 * (dy/dx) + 4y(dy/dx) = (y/x)1/2

[multiply thru by 2 and cancel]

-4x + (x/y)1/2 * (dy/dx) - 4y(dy/dx) = -(y/x)1/2

[multiply thru by -1]

+ 4x                                   +4x  add this to
_______________________________________________ each side

(x/y)1/2 * (dy/dx) - 4y(dy/dx) = 4x - (y/x)1/2  add

(dy/dx)[(x/y)1/2 - 4y] = 4x - (y/x)1/2    factor

dy/dx =      4x - (y/x)1/2         divide each side by this
_________________   and cancel
(x/y)1/2 - 4y

(9.)  (4x2y2)1/5 = 1              here is the problem

4x2y2 = 1          raise each side to the 5th power

x2y2 = .25          divide each side by 4, cancel

xy = 0.5      take the square root of each side

x(dy/dx) + y = 0   take the derivative implicitly

-  y   -y      subtract y from each side
________________________

x(dy/dx)    =  -y     subtract

dy/dx = -y/x       divide each side by x and cancel

(10.)  (1/x2) - (1/y2) = x + y

y2 - x2 = (xy)2(x + y)    multiply thru by x2y2, cancel

(y - x)(y + x) = (xy)2(x + y)     factor

y - x = (xy)2        divide each side by (x + y), cancel

y - x = x2y2           laws of exponents

(dy/dx) - 1 = 2xy2 + 2x2y(dy/dx)  take the derivative

implicitly

+1   +1           add 1 to each side
___________________________________
(dy/dx)   = 1 + 2xy2 + 2x2y(dy/dx)     add

- 2x2y(dy/dx)         - 2x2y(dy/dx)  subtract this from
________________________________________ each side
(dy/dx) - 2x2y(dy/dx) = 1 + 2xy2        subtract

(dy/dx)(1 - 2x2y) = 1 + 2xy2          factor

(dy/dx) = (1 + 2xy2)/(1 - 2x2y)   divide each side by this &

cancel