use implicit differentiation to find dy/dx for this 8xy+x^2=2y^2+3?

Question

please provide a detailed step by step process with solution

Answer 1

Forx:

x=sqrt(72 y^2+12)/2-4 y

OR

x=-sqrt(72 y^2+12)/2-4 y

Fory:

y=-sqrt(72 x^2-24)/4+2 x

OR

y=sqrt(72 x^2-24)/4+2 x

Answer 2

Differentiating the expression:

8 y + 8 x y' + 2 x = 2 y y'

so we have

y' = (8 y + 2 x) /(2 y - 8 x) = (4 y + x) / (y - 4 x)

This is valid as far as nobody said to find dy/dx as a function of x exclusively. If you want y' = g(x), then solve the cuadratic equation and substitute y in the y' expression.

Answer 3

differentiating with respect to x on both sides

(u must use product rule while diff 8xy)


8y + 8x y' + 2x = 4yy'

where y' = dy / dx

y' = 2( y - 2x) / (4y + x)

Answer 4

=8(y+xy´) +2x = 4y*y´ 8y +8xy´+2x =4y*y´

8y+2x= (4y-8x)*y´ so y´= dy/dx= (8y+2x)/(4y-8x) = (4y+x)/(2y-4x)