How To Draw A Tangent To A Curve

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Hi All !

I am new to Mathematica...
Could someone pl. suggest how to draw a tangent line on a curve... a demo would be great
I am working on finding instanteous rate of change of at a point ...

Thanx

Here is a solution. I've picked a function that I am fond of, but you can adapt to your own purposes:

function which takes the name of function of one argument, a position to evaluate the derivative and a parameter that will be used for the tangent.

tangentLine[func_, loc_, param_] :=
Module[
{deriv, x},
deriv = D[func[x], x] /. x -> loc;
deriv (param - loc) + func[loc]
]

For example:

f[t_] := t^2 + Cos[t]
tangentLine[f, 2, z]

Here it is with a function that I have defined within a module, others might have put the function definition in an Initialization, but I think this is easier to read.

Module[
{
   myFunction,
   tangent,
   t
   },
myFunction[0] = 0;
myFunction[1] = 0;
myFunction[x_] := 3 x (1 - x) + (x Log[x] + (1 - x) Log[1 - x]);
Manipulate[
tangent = tangentLine[myFunction, where, t];
Plot[myFunction[z], {z, 0, 1},
Epilog -> {Red,
   Line[{{0, tangent /. t -> 0}, {1, tangent /. t -> 1}}]}
],
{{where, 0.25}, 0, 1}
]
]

Example result:

Here are approaches for single variable: tangent line and two variable functions (tangent plane).

Single variable function:
tg[f_, x_, p_] := (f'[x] /. x -> p) (x - p) + f[p]
Here the argument f is function, x is symbol for differentiation and p is the x-value of point the line will be tangent to.
1. Example using a pure function:
  q = #^3 - 3 #^2 &;
  Manipulate[
  Plot[{q[x], tg[q, u, m] /. u -> x}, {x, -3, 3}, PlotRange -> {-5, 5},
  Epilog -> {Red, PointSize[0.02], Point[{m, q[m]}]}], {m, -1, 1,
  0.01}]
2. Visualizing:
Tangent plane
tangent[f_, arg_,
pt_] := (Grad[f @@ arg, arg] /. Thread[arg -> pt]).(arg - pt) +
f @@ pt

This is essentially a generalization of the previous code exploiting the gradient function and dot product.
1. Example:
  w[x_, y_] := Sin[x] Cos[y];
2. Visualizing:
  ListAnimate[
     Table[
     Show[Plot3D[w[x, y], {x, -3, 3}, {y, -3, 3},
        ViewVector -> {10, 10, 10}, PlotStyle -> Opacity[0.7]],
        Graphics3D[{Red, PointSize[0.03],
        Point[{Join[{j, j}, {Sin[j] Cos[j]}]}]}],
        Plot3D[Evaluate[(tangent[w, {x, y}, {j, j}] /. {x -> u,
           y -> v})], {u, j - 0.5, j + 0.5}, {v, j - 0.5, j + 0.5},
        Mesh -> False, PlotStyle -> Green]], {j, -3, 3, 0.1}]];