Mathematica

 What is the difference between the following two mathematica code?

1. V1func[f_] := Function[x, Evaluate@D[f[x], x]]
2. V2func[f_] := D[f, x] 

Both of the provided lines define the function `V1func` to compute the first derivative of a function \( f \), but they do so in different ways and they will behave differently. Let's examine each one:

1. `V1func[f_] := Function[x, Evaluate@D[f[x], x]]`

   - This creates a new, anonymous function that returns the first derivative of \( f \) when evaluated at a point \( x \).

   - It's a way of generating a function based on another function. 

   - For example, if you were to do:

     ```mathematica

     g = V1func[Sin]

     g[Pi]

     ```

     The output would be the cosine of \( \Pi \) (i.e., -1).

2. `V2func[f_] := D[f, x]`

   - This directly computes the symbolic derivative of the expression \( f \) with respect to \( x \).

   - It does not produce a function. Instead, it produces an expression.

   - If you use this on an expression, it would directly give you the derivative of that expression:

     ```mathematica

     V2func[Sin[x]]

     ```

     The output would be `Cos[x]`.

In summary:

- The first version returns a **function** that can evaluate the derivative at various points \( x \).

  

- The second version computes and returns a **symbolic expression** for the derivative.

The choice between them depends on what you intend to do with the result. If you're going to be evaluating the derivative at various points, the first approach is suitable. If you just want a symbolic expression of the derivative, the second approach is more direct.

Note: V1func[Sin] is a function. V2func[Sin] is nothing. V1func[Sin][x] is derivative of sin at x. V2func[Sin[x]] is also the derivative of sin at x.