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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 $\frac{\partial f(t,s)}{\partial t}$.

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