Implicit Differentiation Help...Please!?

Question

From dy/dx by implicit differentiation and evaluate the derivative at (2, -1).
x2y+y2x=-2

Please help me.

Answer 1

Reading this as:-
x² y + y² x = - 2
2x y + (dy/dx) (x²) + (2y) (dy/dx) x + (1) y² = 0
(x² + 2y) (dy/dx) = - y² - 2xy
dy/dx = (- y) (y + 2x) / (x² + 2y)

Answer 2

- First you differentiate the given function:
x.2dy/dx + 2y + 2y +2x dy/dx =-2
(use product rule to differentiate them)
- Move all "dy/dx" to one side and numbers and y to one side:
2x dy/dx +2y dy/dx = -2 -4y
=> dy/dx (2x+2y) = -2(1+2y)
=> dy/dx = -2(1+2y)/ 2(x+y)
=> dy/dx = (1+2y)/ (x+y)

dy/dx is the derivative so now you only need to substitute the given value (2,-1) into x and y. Therefore we have
(1+2.(-1))/ (2-1)= -1