Question

Find dy/dx by implicit differentiation. (X^2/(x+y))=y^2+1

Answer 1

Apply quotient rule and chain rule and power rule.

Differentiate both sides with respect to x

You'll get

$\frac{2x(x + y) - x^2(1 + y')}{(x+y)^2} = 2yy'\\\\ \frac{2x^2 + 2xy - x^2+ x^2y'}{(x+y)^2} = 2yy'\\\\2x^2 + 2xy - x^2- x^2y' = 2yy'(x+y)^2\\\\ x^2 + 2xy = 2yy'(x+y)^2 + x^2y'\\\\ x^2 + 2xy = y'(2y(x+y)^2 + x^2)\\\\ y' = \boxed{\bf{\frac{x^2 + 2xy}{2y(x+y)^2 + x^2}}}$