[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]
+(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