We compare the meaning of the gradient in the first and last example

Show[ContourPlot3D[{x^y y^z z^x ==1, x+y+z==3}, {x, .5, 3}, {y,.5, 3}, {z, .5, 3}],ParametricPlot3D[{1+t, 1+t, 1+t},{t, -.5, 1}],ListPointPlot3D[{{1, 1, 1}}, Filling->Bottom, PlotStyle->PointSize[Large], PlotStyle->Black]]

Show[ContourPlot3D[{z==10-x^4-x y-y^2,-6(x-1)+-5(y-2)+-(z-3)==0},{x, 0, 2}, {y, 1, 3},{z, -20, 10}],ListPointPlot3D[{{1, 2, 3}}, Filling->Bottom, PlotStyle->PointSize[Large], PlotStyle->Black],ParametricPlot3D[{{1-6 t, 2-5t, 3}}, {t, -2, 2}, ColorFunction->"Rainbow"]]

What is the meaning of the gradient in the second graph?

Really want sphere at origin, but we rotate it so the view is better.

Show[ContourPlot3D[(((y-3))^2)+(x^2)+(z^2)==(3^2), {x, 0, 3.2}, {y, 0, 3.2}, {z, 0, 3.2}], ListPointPlot3D[{{1,1,2}}, Filling->Bottom, PlotStyle->PointSize[Large], PlotStyle->Black]]

Clear[u]

Animate[Show[ContourPlot3D[(((y-3))^2)+(x^2)+(z^2)==(3^2), {x, 0, 3.2}, {y, 0, 3.2}, {z, 0, 3.2}, Mesh->None,ContourStyle->Opacity[0.3]], ListPointPlot3D[{{1,1,2},{1,1,0}}, Filling->Bottom, PlotStyle->PointSize[Large], PlotStyle->Black],ParametricPlot3D[{1+Cos[u]t, 1+ Sin[u] t, 0},{t,0,1}]],{u,0,2 Pi}]

The partial derivatives are directional derivatives in different directions.  The partial with respect to x computes the slope of the green lines when the red vector is pointing in the positive x direction.

f[0]

The partial with respect to x computes the slope of the green line when the red vector points in the positive y direction.

f[Pi/2]

But these are just two directions.  We can define the directional derivative at a point in a direction.

f1[5.8]

We add in the plane that contains the point, sticks straight up, and contains the direction.

f2[5.8]

The intersection of the plane and the graph defines a curve in the plane.  The graph on the right is given by f(r(t)), where r(t) parameterizes the red path with r(0) the original point

f3[5.8]

The directional derivative computes the derivative of the graph on the right.  More specifically, it is the slope of the tangent line in the plane

f[5.8]

Now we can animate this

Animate[f[u],{u, 0, 6Pi}]

f[u_]:=GraphicsRow[{
Show[ContourPlot3D[{(((y-3))^2)+(x^2)+(z^2)==(3^2), -Sin[u](x-1)+Cos[u](y-1)==0}, {x, 0, 3.2}, {y, 0, 3.2}, {z, 0, 3.2}, Mesh->None,ContourStyle->Opacity[0.3]], ListPointPlot3D[{{1,1,2},{1,1,0}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]],ParametricPlot3D[{1+Cos[u]t, 1+ Sin[u] t, 0},{t,0,1},ColorFunction->Function[{x,y,z},Hue[0.04,0.95,1]]], ParametricPlot3D[{1+Cos[u] s, 1+ Sin[u] s, (((9-(((1+Cos[u] s))^2)-(((1+ Sin[u] s-3))^2)))^(1/2))},{s, -1,1}, ColorFunction->Function[{x,y,z},Hue[2/3]]],ParametricPlot3D[{1+Cos[u] s, 1+ Sin[u] s, (((-Cos[u])/2)+Sin[u])s + 2},{s, -1,1}, ColorFunction->Function[{x,y,z},Hue[1/2]]]],
Show[ListPlot[{{1,1}}, ColorFunction->Red],ParametricPlot[{1+Cos[u] t, 1 + Sin[u] t}, {t, 0, 1}, ColorFunction->Function[{x,y},Hue[0.04,0.95,1]]]],
Show[Plot[(((9-(((1+Cos[u] a))^2)-(((1+ Sin[u] a-3))^2)))^(1/2)),{a, -1,1}, ColorFunction->Function[{x,y}, Hue[2/3]]],Plot[ (((-1)/2)Cos[u]+Sin[u]) b + 2, {b,-1,1}, ColorFunction->Function[{x,y}, Hue[1/2]]],ListPlot[{{0,2}}]]
}, ImageSize->Full]

f1[u_]:=GraphicsRow[{
Show[ContourPlot3D[{(((y-3))^2)+(x^2)+(z^2)==(3^2)}, {x, 0, 3.2}, {y, 0, 3.2}, {z, 0, 3.2}, Mesh->None,ContourStyle->Opacity[0.3]], ListPointPlot3D[{{1,1,2},{1,1,0}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]],ParametricPlot3D[{1+Cos[u]t, 1+ Sin[u] t, 0},{t,0,1},ColorFunction->Function[{x,y,z},Hue[0.04,0.95,1]]] ],
Show[ListPlot[{{1,1}}, ColorFunction->Red],ParametricPlot[{1+Cos[u] t, 1 + Sin[u] t}, {t, 0, 1}, ColorFunction->Function[{x,y},Hue[0.04,0.95,1]]]],
Show[ListPlot[{{0,2}}]]
}, ImageSize->Full]

f2[u_]:=GraphicsRow[{
Show[ContourPlot3D[{(((y-3))^2)+(x^2)+(z^2)==(3^2), -Sin[u](x-1)+Cos[u](y-1)==0}, {x, 0, 3.2}, {y, 0, 3.2}, {z, 0, 3.2}, Mesh->None,ContourStyle->Opacity[0.3]], ListPointPlot3D[{{1,1,2},{1,1,0}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]],ParametricPlot3D[{1+Cos[u]t, 1+ Sin[u] t, 0},{t,0,1},ColorFunction->Function[{x,y,z},Hue[0.04,0.95,1]]] ],
Show[ListPlot[{{1,1}}, ColorFunction->Red],ParametricPlot[{1+Cos[u] t, 1 + Sin[u] t}, {t, 0, 1}, ColorFunction->Function[{x,y},Hue[0.04,0.95,1]]]],
Show[ListPlot[{{0,2}}]]
}, ImageSize->Full]

f3[u_]:=GraphicsRow[{
Show[ContourPlot3D[{(((y-3))^2)+(x^2)+(z^2)==(3^2), -Sin[u](x-1)+Cos[u](y-1)==0}, {x, 0, 3.2}, {y, 0, 3.2}, {z, 0, 3.2}, Mesh->None,ContourStyle->Opacity[0.3]], ListPointPlot3D[{{1,1,2},{1,1,0}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]],ParametricPlot3D[{1+Cos[u] s, 1+ Sin[u] s, (((9-(((1+Cos[u] s))^2)-(((1+ Sin[u] s-3))^2)))^(1/2))},{s, -1,1}, ColorFunction->Function[{x,y,z},Hue[2/3]]],ParametricPlot3D[{1+Cos[u]t, 1+ Sin[u] t, 0},{t,0,1},ColorFunction->Function[{x,y,z},Hue[0.04,0.95,1]]] ],
Show[ListPlot[{{1,1}}, ColorFunction->Red],ParametricPlot[{1+Cos[u] t, 1 + Sin[u] t}, {t, 0, 1}, ColorFunction->Function[{x,y},Hue[0.04,0.95,1]]]],
Show[Plot[(((9-(((1+Cos[u] a))^2)-(((1+ Sin[u] a-3))^2)))^(1/2)),{a, -1,1}, ColorFunction->Function[{x,y}, Hue[2/3]]],ListPlot[{{0,2}}]]
}, ImageSize->Full]