Question 1045479
<pre>
The equation of the tangent plane to the surface 
{{{"F(x,y,z)"}}}{{{""=""}}}{{{0}}} at the point {{{(matrix(1,5,x[0],",",y[0],",",z[0])))}}}

is

{{{F[x]}}}{{{(matrix(1,5,x[0],",",y[0],",",z[0]))*(x-x[0]))}}}{{{""+""}}}{{{F[y]}}}{{{(matrix(1,5,x[0],",",y[0],",",z[0]))*(y-y[0]))}}}{{{""+""}}}{{{F[z]}}}{{{(matrix(1,5,x[0],",",y[0],",",z[0]))*(z-z[0]))}}} {{{""=""}}} {{{0}}}

{{{x^2 + y^2 + z^2}}}{{{""=""}}}{{{1}}}
 
{{{x^2 + y^2 + z^2-1}}}{{{""=""}}}{{{0}}}

{{{"F(x,y,z)"}}}{{{""=""}}}{{{x^2 + y^2 + z^2-1}}}

{{{F[x]}}}{{{"(x,y,z)"}}}{{{""=""}}}{{{2x}}}

{{{F[y]}}}{{{"(x,y,z)"}}}{{{""=""}}}{{{2y}}}

{{{F[z]}}}{{{"(x,y,z)"}}}{{{""=""}}}{{{2z}}}

{{{F[x]}}}{{{(matrix(1,5,1/2,",",1/2,",",sqrt(2)/2)))}}}{{{""=""}}}{{{2(1/2)}}}{{{""=""}}}{{{1}}}

{{{F[y]}}}{{{(matrix(1,5,1/2,",",1/2,",",sqrt(2)/2)))}}}{{{""=""}}}{{{2(1/2)}}}{{{""=""}}}{{{1}}}

{{{F[z]}}}{{{(matrix(1,5,1/2,",",1/2,",",sqrt(2)/2)))}}}{{{""=""}}}{{{2(sqrt(2)/2)}}}{{{""=""}}}{{{sqrt(2)}}}


So the equation of the tangent plane at {{{(matrix(1,5,1/2,",",1/2,",",sqrt(2)/2)))}}} is

{{{1(x-1/2)+1(y-1/2)+sqrt(2)(z-sqrt(2)/2)}}}{{{""=""}}}{{{0}}} 

{{{x-1/2+y-1/2+sqrt(2)z-2/2}}}{{{""=""}}}{{{0}}} 

{{{x-1/2+y-1/2+sqrt(2)z-1}}}{{{""=""}}}{{{0}}} 

{{{x+y+sqrt(2)z-2}}}{{{""=""}}}{{{0}}}

Edwin</pre>