Using implicit differentiation find dy/dx at point (-1,-2) if...

Question

-4x^3 + y^3 + y^2 = 0.

Answer 1

Using implicit differentiation find dy/dx at point: (x, y) = (-1, -2) if... -4x³ + y³ + y² = 0.

Detailed Solution:

-4x³ + y³ + y² = 0 ==> d/dx(-4x³ + y³ + y² = 0) ==> d/dx(-4x³) + d/dx((y³ ) + d/dx((y²) = d/dx((0) ==>

-12x² + 3y²dy/dx + 2ydy/dx = 0 ==> 3y²dy/dx + 2ydy/dx = 12x² ==> (3y² + 2y)dy/dx = 12x² ==>

dy/dx = (12x²)/(3y² + 2y)

dy/dx_{(-1, -2)} = (12(-1)²)/(3(-2)² + 2(-2)) = 12/(12 – 4) = 12/8 = 3/2 ==> dy/dx = 3/2

Answer 2

To make things a little nicer, rewrite equation as y³+y² = 4x³. Then taking the derivative of both sides with respect to x,

12x² = 3y²(dy/dx) + 2y(dy/dx) = (3y²+2y)(dy/dx)

and solving for dy/dx

dy/dx = 12x²/(3y²+2y).

Final step is to evaluate dy/dx at (-1,-2).