Suppose we have a simple Ito diffusion model
$$dX(t)=\mu dt + \sigma dW(t)$$
and function which is at least twice differentiable $y=k(x)$
then by Ito's Lemma we have $$ {\mathrm{d} k(X(t))} = \left(\frac{\partial k(X(t))}{\partial t} + \mu \frac{\partial k(X(t))}{\partial x} + \frac{\sigma^2}{2} \frac{\partial^2 k(X(t))}{\partial x^2}\right)dt + \sigma \frac{\partial k(X(t))}{\partial x} dW(t) $$
However, suppose we are in a setting where the function $k(x)$ is unknown and we want to separate it somehow. Is the following "classical chain rule" allowed?
$$ \frac{\partial k(X(t))}{\partial t} = \frac{\partial k(x)}{\partial x} \frac{\partial k(X(t))}{\partial t} $$
I am interested on how we approach situations where we know $X(t)$ and we know the integrated process of the transformed $X(t)$ by a function i.e.: $k(x)$ - then is it possible to solve or approximate the function...?