Derivative of a function w.r.t another function

Question

I have the function $f(X)$, $X$: depend to a variable $\theta $. is there a direct expression to calculate the derivative of $f(X)$ with respect to $\theta$:

$$\frac{df(X)}{d\theta} $$

And I want to provide me with some references if you would.

Answer 1

According to the chain rule:

$$\frac{\partial f}{\partial \theta} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial \theta},$$

that is, you first differentiate the outer function $f$ wrt. to the inner function $x$, and then differentiate the inner function wrt. to the variable you're originally differentiating wrt. (here $\theta$), and then you multiply them to get the desired result.

EDIT, as response to OPs edit:

It seems you are looking for the so-called total derivative, where a hard d is used:

$$\frac{\mathrm{d} f}{\mathrm{d} \theta} = \frac{\partial f}{\partial \theta}+\frac{\partial f}{\partial x} \frac{\partial x}{\partial \theta}.$$

Note that $$\frac{\mathrm{d} x}{\mathrm{d} \theta}=\frac{\partial x}{\partial \theta},$$ as $x$ only depends on $\theta$ (at least, this is assumed from the information given).