I'm having trouble understanding applying chain rule to SDEs or actually chain rules in general. It has been a while since I took rudimentary calculus classes, so I might be slipping on the basics.
Suppose there is an SDE that follows the dynamic $$ dX(t) = \alpha X(t) dt + \sigma X(t) dW(t) $$ Lets take a function $ f(X(t)) $ and take the derivative. Applying Ito's formula, the derivative should have the following expression $$ df = f_t dt + f_x dx + \frac{1}{2} f_{xx} (dx)^2$$ $$ = \frac{\partial}{\partial t} f dt + \frac{\partial}{\partial x } dx + \frac{\partial^2}{\partial x^2} f (dx)^2 $$ So the last two terms I understand, but it is the first term that I am having a hard time comprehending. Since $X(t)$ is a function of $t$, I believe it should be $$ \frac{\partial f(X(t))}{\partial t} dt = \frac{\partial f(X(t))}{\partial X(t)} \frac{\partial X(t)}{\partial t} dt $$ But, in reality, this term is set to zero, which is something that I cannot understand. Is this term in fact zero? If so, what is it that I'm missing?
For example, if $f:=e^{X(t)}$, then $$ de^{X(t)} = e^{X(t)} dX(t) + \frac{1}{2} e^{X(t)} (dX(t))^2 $$
Partial derivatives ignore implicit dependence. $\partial/\partial t f(x)=0$ since there is no explicit $t$ dependence, but $d/dt f$ might not be zero.
In your last equation, you have done the actual derivative, not the partial derivatives with respect to $t$.