This device is known as the gradient. The gradient equals (derivative of 1st component w/ respect to x, derivative of 2nd component w/ respect to y, ..., nth component w/ respect to nth variable).
The gradient is the perpendicular or normal vector to the surface at a certain point. How does this relate to tangent planes? Well, for implicit surfaces, the tangent plane is the set of points (x,y,z) that satisfy the equation (grad f(a,b,c))((x,y,z)-(a,b,c)) = 0 where (a,b,c) is a specific point. (This means that the gradient is, at all times, perpendicular to our tangent plane. So, to get our tangent plane, we simply derive the plane perpendicular to our gradient vector.)
Let us define a function f, f: x^2+ y^2+ z^2=2. The point that we have interest in is (a,b,c)=(1,1,0). grad f = (2x, 2y, 2z)
grad f(a,b,c) = (2,2,0)
(x,y,z)-(a,b,c) = (x-1, y-1, z)
The resulting formula of our tangent plane to f at (a,b,c) is (x-1, y-1, z).(2,2,0) =0 Note that the third component of the plane is given free range.
A. We define our function implicitly as q:= z^3+y^2+x=4. Our point of interest is (2,1,1).
B. Find the gradient vector to q.
C. Plug in your point of interest.
D. After you have this gradient, find the plane perpendicular to this vector, and voila, you have your tangent plane.
E. Use the spiffy Maple tool to check your work. (If you have not already pasted in the maple code that auxilliary functions, then do so now.) The syntax is as follows:
ImplicitTan(function, x=xmin..xmax, y=ymin..ymax, z=zmin..zmax, point =[x0,y0,z0]);
Note: Remember that when using ImplicitTan on your own that the point you specify must lie on the surface, otherwize you will not get tangent planes.