Let $f_i$ with $i=1,2,3$ be $C^1$ function from $\mathbb{R}$ to $\mathbb{R}$. I want to compute the derivative of$$h(x) = (|\max(f_1(x),f_2(x), f_3(x))|-x^2)^2.$$so$$h'(x) = 2 g(x)(\partial_x |\max(f_1(x),f_2(x), f_3(x))| -2x)$$and since $\partial |x|= \frac{x}{|x|}= \mathrm{sign}(x)$ we then have$$g'(x) =2 g(x)( \max(f_1'(x),f_2'(x), f_3'(x))\frac{|\max(f_1(x),f_2(x), f_3(x))|}{\max(f_1(x),f_2(x), f_3(x))} -2x).$$is this derivation is true?
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