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 (x^{2} + 3xy + y^{4}) = d/dx (x^{2} + 22)

2x + 3y + 3x*dy/dx +4y^{3}*dy/dx = 2x

Now, put all the dy/dx terms on one side of the equation

dy/dx(3x+4y^{3}) = 3y

dy/dx = 3y/(3x+4y^{3})

Use the point (1,2) and plug it in to solve for dy/dx

dy/dx = 3*2/(3*1 + 4*2^{3}) = **6/35**