I have a very complicated function for which I need to evaluate
$$ \frac{dF(x)}{dx}\Bigg|_{x-x_0}.$$
How can I tell when this will be equivalent to evaluating
$$ \frac{dF(x-x_0)}{dx}\Bigg|_x ?$$
Under what conditions on $F$ is this equivalent? Would this condition change for higher derivatives?
Assuming $x_0$ is a constant, the Chain Rule gives $\frac{dF(x-x_0)}{dx}\Bigg|_x =\frac{dF(x-x_0)}{dx}=F'(x-x_0)\cdot \frac{d(x-x_0)}{dx} =\frac{dF(x)}{dx}\Bigg|_{x-x_0}$
Assuming $g(x) := F(x-x_{0})$ is well-defined and differentiable, we have: $$\frac{dF(x)}{dx}\bigg{|}_{x-x_{0}} = \lim_{y \to 0}\frac{F(y+(x-x_{0})) - F(x-x_{0})}{y} = \lim_{y\to 0} \frac{F((y+x)-x_{0})-F(x-x_{0})}{y} = \lim_{y\to 0}\frac{g(y+x)-g(x)}{y} = \frac{dg(x)}{dx} = \frac{dF(x-x_{0})}{dx}\bigg{|}_{x}$$
