Let $C := \{ g: [a,b] \times \mathbb{R} \rightarrow \mathbb{R} | a, b \in \mathbb{R}, a < b \text{ and } g \text{ is derivable} \} $
Let $f: [0,\infty[ \times \mathbb{R} \rightarrow \mathbb{R}: (z,u) \mapsto f(z,u)$ a derivable function in all variables.
Let $g: [\Delta t, \infty[ \rightarrow C: t \mapsto g_t$ with $g_t: [t-\Delta t, t+\Delta t] \times \mathbb{R} \rightarrow \mathbb{R}: (t',u) \mapsto f(t',u)$
How can we calculate $\dfrac{\partial g}{\partial t}$ ?
I don't have a clue how to calculate this. Can someone help me ?