What, *exactly*, do you want to illustrate? Just a single tangent line to the curve with a point moving along the curve? (or that but with the point moving along the tangent?) Or do you want a moving tangent line as you move points along the curve?

Whichever you want, there are some problems with your code. Before using `Manipulate`

, it's a good idea to try the graphics you want in a static mode, with a fixed value for the control variable(s).

makes no sense to me: on the left-hand side you seem to have two arguments, the first being `a`

; but what is the second argument? you have `1/2 _`

, with a space between the fraction and the pattern symbol `_`

; what is that supposed to mean? Moreover, you have an `x`

on the right-hand side but no `x`

as an argument.

your `Plot`

code

Plot[ {y[x, 1/2], f[x]}, {x, -2, 4}, {1/2, -2, 4} (* options *)]

makes no sense: you seem to want to plot two functions of variable `x`

, namely, `y[x, 1/2]`

and `f[x]`

, but then after the domain list `{x, -2, 4}`

(which is fine) you have the mysterious list `{1/2, -2, 4}`

.

Once you straighten all that out, you can try to put it inside the `Manipulate`

.

I also suggest that you *not* directly use `f'[a]`

but rather compute ahead of time, once and for all, the derivative functions, say:

fp[x_] = f'[x]

And then use that in the right-hand side

f[a] + fp[a] (x - 1/2)

(or whatever it is that you *really* are trying to get there).

Note also that there is no earthly reason to include the explicit `*`

symbol for multiplication in `f[a] + f'[a] * (x - 1/2)`

or similar symbolic expression; explicit multiplication signs are needed in *Mathematica* only between actual numbers.