I want to apply the following formula
$$dV(t,w(t)) = \left(\frac{\partial}{\partial t}V(t,w(t)) + \frac{1}{2} \frac{\partial^2}{\partial x^2}V(t,w(t))\right)dt + \frac{\partial}{\partial x}V(t,w(t))dw(t). $$
to find the differentials of the following functions:
- $V(t,w(t)) = \sin(w(t)) $
- $V(t,w(t)) = \exp(t^2-2w^2(t))$
In both cases, $w(t)$ is a Wiener process.
I am a little bit confused because I have derivatives with respect to $x$ in the formula and the functions I'm given has no $x$'s in them. The examples if the book starts with $V(t,x)$. Then to find for example $\frac{\partial}{\partial x}V(t,w(t))$, I evaluate the partial derivative of $V$ wrt $X$ at $w(t)$.
I would like some help in starting off.