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


`(2x(x + y) - x^2(1 + y'))/((x+y)^2) = 2yy'\\\\ (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{(x^2 + 2xy)/(2y(x+y)^2 + x^2)}}`