find dy/dx

Question

1) x² - xy + x + y -1 = 0

2) e^xy + x + y = 1

3) logx + logy = x - y

4) x³ + y³ + x²y + y² = 0

Answer 1

(1) 2x-xdy/dx-y+1+dy/dx=0,

-xdy/dx+dy/dx=-2x+y-1,

(dy/dx)(-x+1)=-2x+y-1,

dy/dx=(y-2x-1)/(1-x) or (2x-y+1)/(x-1).

(2) (a) (d/dx)(e^xy+x+y-1)=e^xdy/dx+e^xy+1+dy/dx=0, dy/dx=-(e^xy+1)/(e^x+1);

(b) (d/dx)(e^xy+x+y-1)=e^xy(y+xdy/dx)+1+dy/dx=0, dy/dx=-(e^xyy+1)/(xe^xy+1).

(3) 1/x+(1/y)(dy/dx)=1-dy/dx, dy/dx(1+1/y)=1-1/x,

dy/dx=(1-1/x)/(1+1/y)=((x-1)/x)(y/(y+1))=y(x-1)/(x(y+1)).

(4) 3x²+3y²dy/dx+2xy+x²dy/dx+2ydy/dx, dy/dx=-(3x²+2xy)/(3y²+x²+2y).