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Consider the function $F:\mathbb{R}\rightarrow \mathbb{R}$.

Let $f(x)$ denote the derivative of $F$ at $x\in \mathbb{R}$.

Consider two other differentiable functions $g:\mathbb{R}\rightarrow \mathbb{R}$ and $b:\mathbb{R}\rightarrow \mathbb{R}$.

Suppose I want to compute $$ \frac{\partial}{\partial x} F\Big(\min\Big\{g(x), b(x)\Big\}\Big). $$ Assume that $g(x)\neq b(x)$ for each $x\in \mathbb{R}$.

Is it correct that $$ \frac{\partial}{\partial x} F\Big(\min\Big\{g(x), b(x)\Big\}\Big)=\mathbb{1}_{[g(x)< b(x)]} f(g(x)) \frac{\partial g(x)}{\partial x}+\mathbb{1}_{[g(x)> b(x)]} f(b(x)) \frac{\partial b(x)}{\partial x} \quad ? $$

Star
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  • If $g(x) \ne b(x)$ for each $x$ in $\mathbb R$, then either one of them is discontinuous or one of them is always larger than the other. – eyeballfrog Feb 03 '23 at 17:47
  • But since we're assume $g$ and $b$ are differentiable on all of $\Bbb R$, they must be everywhere continuous. A few more comments: Please do not write partial derivative $\partial/\partial x$ when you are dealing with functions of a single variable. (You also forgot half the derivative symbol twice on the right.) – Ted Shifrin Feb 03 '23 at 17:51
  • Thanks. Is the answer to my question"yes"? – Star Feb 03 '23 at 17:58

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