How do I find the equation of a tangent plane of a function at a point in 3D?

If given f(x,y), I'm asked to find the equation of the tangent plane at (x,y,z). Do I find df/dx and df/dy and only use the x and y coordinates? How about the z coordinate?How do I find the equation of a tangent plane of a function at a point in 3D?If z=f(x,y), then dz = (df/dx) dx + (df/dy) dy .

Therefore, at the point (x0, y0, z0) the equation of the tangent plane is

(df/dx) (x-x0) + df/dy (y-y0) - (z-z0) = 0 where the derivatives are to be evaluated at the point (x0,y0) too.



So here you see how the z-coodinate of the tangent point (z0) needs to be used.



Example: z = 4 - x^2 - y^2. Equation of the tangent plane though (1,1,2)?



df/dx = -2 x

df/dy=-2 y



So in (1,1,2)



df/dx = -2

df/dy = -2



-2 (x - 1) - 2( y - 1) - (z - 2) = 0



or



2 x + 2 y + z = -6