What is the derivative of max(u(x),v(x)) when u(x) and v(x) are given functions?

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The derivative of the function f(x) = max(u(x), v(x)) depends on the relationship between the two functions u(x) and v(x). Specifically, the derivative can be determined by analyzing the cases where u(x) > v(x) and v(x) > u(x). Notably, f(x) can be differentiable even when u(x) and v(x) are not continuous, as demonstrated by the examples where u(x) = 0 for rational x and u(x) = 1 otherwise, and vice versa for v(x). Additionally, there are scenarios where f(x) is not differentiable, such as when u(x) = x and v(x) = -x, resulting in f(x) = |x|, which is not differentiable at x = 0.

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What is the derivative of the function f(x)= max(u(x),v(x)) ?
where u(x) and v(x) are two given function
 
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Try looking at the two cases when u(x)>v(x) and when v(x)>u(x)
 
Office_Shredder said:
Try looking at the two cases when u(x)>v(x) and when v(x)>u(x)

That is a good place to start, but max(u(x),v(x)) can be differentiable when u(x) and v(x) are not even continuous.

For example
u(x) = 0 when x is rational, u(x) = 1 otherwise
v(x) = 1 when x is rational , v(x) = 0 otherwise
 
AlephZero said:
That is a good place to start, but max(u(x),v(x)) can be differentiable when u(x) and v(x) are not even continuous.

For example
u(x) = 0 when x is rational, u(x) = 1 otherwise
v(x) = 1 when x is rational , v(x) = 0 otherwise

Or max(u(x),v(x)) can not be differentiable, while u(x) and v(x) are:

For example:
u(x)=x and v(x)=-x

Then max(u(x),v(x))=|x| which is not differentiable in 0.
 

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