Given a sphere $(x-2)^2+(y-1)^2+(z+1)^2=9$, steps to find the equation of the tangent plane to this sphere at point (1,-1,1) are as follows:
$$(x-2)^2+(y-1)^2+(z+1)^2=9$$ $$(x-2)^2+(y-1)^2+(z+1)^2-9=0$$ $$(x-2)^2+(y-1)^2+(z+1)^2-9=f(x,y,z)$$
$$\nabla f(1,-1,1)=(-2,-4,4) $$ $$f_x(1,-1,1)(x-1)+f_y(1,-1,1)(y+1)+f_z(1,-1,1)(z-1)=0$$
When setting the equation of the sphere to 0, then setting the equation equal to a fourth variable, $f$ is not a sphere anymore.
If the sphere equation is altered in this way, how does the partial derivatives of $f$ associate back to the sphere in that they are apart of the tangent plane equation for the sphere???
The equation of the tangent plane is: $$-2(x-1)-4(y+1)+4(z+1)=0 \implies x+2y-2z=1.$$