sliding a tangent line along a curve

LocatorPane[Dynamic[pt, (pt = {#[[1]], Sin[#[[1]]]}) &],
Dynamic@Plot[{Sin[x],
ConditionalExpression[Cos[pt[[1]]]*(x - pt[[1]]) + Sin[pt[[1]]],
 pt[[1]] - 1 <= x <= pt[[1]] + 1]}, {x, 0, 10},
PlotRange -> {{-2, 11}, {-2, 2}},
Epilog -> {PointSize[Large], Point[pt]}],
Appearance -> None]

For arbitrary function func, replace Cos with Derivative[1][func] in the first argument of ConditionalExpression.

Instead of Epilog -> {PointSize[Large], Point[pt]}], one can also use

Mesh -> {{pt[[1]]}}, MeshStyle -> PointSize[Large]]

Update: Using Locator as a graphics primitive, one can get the same result without having to use LocatorPane:

Dynamic@Plot[{Sin[x],
ConditionalExpression[Cos[pt[[1]]]*(x - pt[[1]]) + Sin[pt[[1]]],
pt[[1]] - 1 <= x <= pt[[1]] + 1]}, {x, 0, 10},
PlotRange -> {{-2, 11}, {-2, 2}},
Epilog ->  Dynamic@{Locator[Dynamic[pt, (pt = {#[[1]], Sin[#[[1]]]}) &],
  Graphics[{Black, PointSize[Large], Point[pt]}]]}]

Even if this is already answered... here a short, straightforward, "solution", which might be instructive to people, not so deeply concerned with all that "Dynamic" stuff ;-)

f[x_] := Sin[x]

tangent[f_, x0_, x_] := f'[x0] (x - x0) + f[x0]

   Manipulate[Plot[{f[x], tangent[f, p, x]}, {x, 0, 2 π},PlotRange -> {{0, 2 π}, {-3, 3}},
Epilog -> {Red, PointSize[.015], Point@{p, Sin[p]}}], {p, 0,  2 π}]

---

(edit by J. M.)

This is a variation of mgamer's proposal; I did not feel it deserved its own answer, but it was too long for a comment. This one eliminates the slider, and uses a Locator instead, so that one can slide the tangent line along the curve:

Manipulate[Plot[f[x], {x, 0, 2 π}, PlotRange -> {{0, 2 π}, {-3, 3}}, 
                Epilog -> {{Directive[ColorData[97, 2], AbsoluteThickness[1]], 
                            InfiniteLine[{p[[1]], f[p[[1]]]}, {1, f'[p[[1]]]}]},
                           {Directive[Red, AbsolutePointSize[6]], 
                            Point[{p[[1]], f[p[[1]]]}]}}],
           {{p, {0, 0}}, Locator, Appearance -> None},
           Initialization :> (f[x_] := Sin[x])]