[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

x^1/2 + y^1/2 = 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]

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

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

dy/dx = -(xy)^1/2 multiply each side by y^1/2, 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

y^15/8 + x^15/8 (dy/dx) = 0 multiply thru by (xy)^15/8, cancel

-y^15/8 -y^15/8 subtract this fr ea side
______________________________
x^15/8 (dy/dx) = -y^15/8 subtract

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

(6.) x^3/4 + y^3/4 = 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

y^1/4 + x^1/4 * (dy/dx) = 0 multiply thru by x^1/4 * y^1/4

-y^1/4 - y^1/4 subt this fr ea side
___________________________________
x^1/4 * (dy/dx) = -y^1/4 subtract

dy/dx = -(y/x)^1/4 div ea side by x^1/4, 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 - x^1/2y^1/2 + y² = 6 use the 1/2 power for radical

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

[take the derivative implicitly]

+(1/2)(y/x)^1/2 + (1/2)(y/x)^1/2 add
_____________________________________________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