Is Derivative Notation Interchangeable When Evaluating at a Point?

[Pacopag]

Homework Statement

Suppose you have a function f(x).
You take the derivative with respect to x and evaluate it at some point x_0.
i.e. {{df(x_{0})}\over{dx}} (e.g. first coeff in taylor series)
Is this the same as changing the argument of f to x_0, i.e. writing f(x_0),
then just taking the derivative with respect to x_0?
In fewer words, can I write
{{df(x_{0})}\over{dx}}={{df(x_{0})}\over{dx_0}}.

Homework Equations

The Attempt at a Solution

If I just consider a bunch of examples, it always seems to be true. It seems like a trivial fact, but I would just like someone to confirm this for me.

Thanks.

 

[jhicks]

No,

\frac{df(x_0)}{dx} may be read as the derivative of f(x) with respect to x, evaluated at x = x_0. I am confused as to how you were taking the derivative of something with respect to a constant.

 

[Pacopag]

You're right. It seems highly unorthodox.
Ok. Consider the taylor series (keeping only first order)
q(\Delta t)=q(0)+{{\partial H(q,p)}\over{\partial p}}\Delta t
p(\Delta t)=p(0)-{{\partial H(q,p)}\over{\partial q}}\Delta t
where the derivatives are evaluated at t=0. I used hamilton's equation.
{{dq}\over{dt}}={{\partial H}\over{\partial p}}
{{dp}\over{dt}}=-{{\partial H}\over{\partial q}}
Now regard this as a transformation from the old coords q(0), p(0) to the new coords q(Delta t), p(Delta t).
What is the Jacobian for this tranformation?