Given a function $\varphi(t)\in C^1(R)$. For $a>0, t\geq 0$ and $s\in[t-a,t]$, we set $f(t,s)=\sup_{s\leq \xi\leq t}\varphi(\xi)$ How to formulate or estimate the derivative $\frac{\partial f(t,s)}{\partial t}$.
Derivative estimation?
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real-analysis