implicit diff.

[Guest]

My prob is to solve x^2 + 3xy + y^2 = 6y by implicit differentiation.

Im having trouble understanding how to diff the function with respect to x & y. Let me just show you:

ok first I diff each term w/respect to x:

d/dx(x^2) + d/dx(3xy) + d/dx(y^2) = d/dx(6y),

so now, diff func of x w/respect to x and fcn of y w/respect to x so:

3x + d/dy(3xy) * dy/dx + d/dy(y^2) * dy/dx = d/dy(6y) * dy/dx

so,
3x + 3x dy/dx + 2y dy/dx = 6 dy/dx, then I rearranged to collect like terms,

3x = 6dy/dx - 3x dy/dx - 2y dy/dx, so,

3x = (6-3x-2y)dy/dx, = dy/dx = 3x/(6-3x-2y)????

 

[galactus]

You forgot the 2x and you forgot the differential with respect to x.

Since you have 3xy, you need (3x)dy/dx and 3y.

Like so:

2x+3y+(3x)dy/dx+(2y)dy/dx-(6)dy/dx=0

(3x)dy/dx+(2y)dy/dx-(6)dy/dx=-2x-3y

(3x+2y-6)dy/dx=-2x-3y

dy/dx=(-2x-3y)/(3x+2y-6)

dy/dx=-(2x+3y)/(3x+2(y-3))