Find the partial derivative of the surface's equation with respect to x, y and z. That involves finding the derivative for each variable while treating the other two as constants. For example, given the equation z^2 - 2zx - 3y + 4xy + x^2 = 2, the partial derivatives fx, fy and fz are:
fx = -2z + 4y + 2x
fy = 3 + 4x
fz = 2z - 2x
Evaluate each of the partial derivatives for the coordinate values of the point of tangency. For example, if you want to find the equation of the tangent plane at the point (1, -2, 3), evaluate the partial derivatives for x = 1, y = -2 and z = 3:
fx = -2*3 + 4*(-2) + 2*1 = -12
fy = 3 + 4x = 7
fz = 2z - 2x = 4
Substitute the values for each partial derivative and each coordinate value into the function fx*(x - x1) + fy*(y - y1) + fz*(z - z1) = 0, where (x1, y1, z1) is the point of tangency. The resulting equation is the tangent plane. For example, -12(x - 1) + 7(y + 2) + 4(z - 3) = 0 simplifies to 12x - 7y - 4z = 14.