Let's use implicit differentiation to solve this. Differentiate each side of the equation with respect to x. Be sure to not forget product rule for the 3xy term.
d/dx (x2 + 3xy + y4) = d/dx (x2 + 22)
2x + 3y + 3x*dy/dx +4y3*dy/dx = 2x
Now, put all the dy/dx terms on one side of the equation
dy/dx(3x+4y3) = 3y
dy/dx = 3y/(3x+4y3)
Use the point (1,2) and plug it in to solve for dy/dx
dy/dx = 3*2/(3*1 + 4*23) = 6/35