Expert: Ahmed Salami - 10/21/2009

Question
Kindly assist with the below:

a) A curve follows the implicit function
x^3+3x^3y^2-2y^4 = 2x+3y

(i) Find its derivative with respect to x (i.e.)dy/dx

Answer
Hi Timothy,
We differentiate every term accordingly but always remembering to add dy/dx whenever we differentiate a term in y.
x^3 + 3x^3.y^2 - 2y^4 = 2x + 3y
differentiating through,
3x^2 + 3[x^3.2y(dy/dx) + 3x^2.y^2] - 8y^3(dy/dx) = 2 + 3(dy/dx)
solving for dy/dx,
3x^2 + 6x^3.y(dy/dx) + 9x^2.y^2 - 8y^3(dy/dx) = 2 + 3(dy/dx)
6x^3.y(dy/dx) - 8y^3(dy/dx) - 3(dy/dx) = 2 - 3x^2 - 9x^2.y^2
(dy/dx)(6x^3.y - 8y^3 - 3) = 2 - 3x^2 - 9x^2.y^2
(dy/dx) = (2 - 3x^2 - 9x^2.y^2)/(6x^3.y - 8y^3 - 3)

Regards