implicit differentiation

Question

Assume that F is differential function with F'(2)=5. Find dy/dx at the point (x,y) = (1,2) if y is defined implicitly by the equation : F(y/x) = y^2

 

Note that z, y, F are not related except through this equation. in particular y =/= F(x)

Answer 1

F(y/x) = y² and F'(2) = 5

 

By the Chain Rule, F'(y/x) [ d/dx(y/x) ] = 2y(dy/dx)

 

                            F'(y/x)[{x(dy/dx) - (1)(y)} / x²] = 2y(dy/dx)

 

So, at the point (1,2), we have F'(2)[{dy/dx - 2} = 4(dy/dx)

 

                                                  5(dy/dx - 2) = 4(dy/dx)

 

                                                              dy/dx = 10