2yx^2 - 7y^2 - 8x + 2y = -11

-(d/dx)/(d/dy)

(8 - 4xy) / (2x^2 - 14y + 2)

y'(1) just accounts for one of those variables, however. You need an x and a y coordinate to perform the calculation. For y'(1), do you mean x by that 1? If so, just plug back into the original equation to get the y value accompanying it.

Edit: Missed that 2y', sorry. Fixed