Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point

x^2 + 2xy - y^2 + x =2 (1,2) (hyperbola)

Answer 1

differentiating wrt \(x\) gives by the chain rule and product rule
\[2x+2(y+xy')-2yy'+1=0\] solve for \(y'\)

Answer 2

or to make the algebra easier, start with \(2x+2(y+xy')-2yy'+1=0\) replace \(x\) by \(1\) and \(y\) by \(2\) and then solve for \(y'\)