I'm trying to derive an HJB from a discrete time setting. At some point, I am left with
$$ \lim_{\Delta\to 0} \frac{v(c_{t+\Delta}, u_{t+\Delta}, t+\Delta) - v(c_{t}, u_{t}, t)}{\Delta}$$
and am not sure what to do. If $\Delta$ was only in one argument, this would be a partial differential. My hunch is that this is the total derivative in $t$, but I dont know how to show that.
How do I proceed with the expression above?
You can separate your function in three terms by writing \begin{align} & v(c_{t+\Delta},u_{t+\Delta},t+\Delta)-v(c_t,u_t,t) = \\ & v(c_{t+\Delta},u_{t+\Delta},t+\Delta)-v(c_t,u_{t+\Delta},t+\Delta) \\ + & v(c_{t},u_{t+\Delta},t+\Delta)-v(c_t,u_t,t+\Delta) \\ + & v(c_t,u_t,t+\Delta)-v(c_t,u_t,t) \end{align} When you divide by $\Delta$ and take the limit $\Delta \rightarrow 0$, the first expression converges to $\dfrac{\partial v}{\partial c} \dfrac{dc}{dt}$, the second expression to $\dfrac{\partial v}{\partial u} \dfrac{du}{dt}$, and the third expression to $\dfrac{\partial v}{\partial t}$. Therefore your total derivative equals \begin{equation} \dfrac{\partial v}{\partial c} \dfrac{dc}{dt} + \dfrac{\partial v}{\partial u} \dfrac{du}{dt} + \dfrac{\partial v}{\partial t} \end{equation}
