Find the equation of the tangent plane to f(x,y) = x^3 - 3xy^2 at
the point (2,3).

Answer: this is of the form z = f(x,y) so that the equation is given by

z = f(a,b) + f_x(a,b) (x - a) + f_y(a,b) (y - b)

(a, b) = (2, 3)
f(2,3) = 2^3 - 3*2*3^2 = 8 - 54 = -46
f_x(x,y) = 3x^2 -3y^2
f_x(2,3) = 3*2^2 -3*3^2 = 12 - 27 = -15
f_y(x,y) = - 6xy 
f_y(2,3) = - 6*2*3 = -36

so the equation is 
z = - 46 - 15 (x - 2) - 36 (y - 3) <--- this is enough



alternate answer
15 x + 36 y + z = 92
check the point (2, 3, -36)
15*2 + 36*3 - 46 = 92?
 30  + 108 - 46 = 92?
             92 = 92 yes.