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

[Dansuer]

What is the derivative of the function f(x)= max(u(x),v(x)) ?
where u(x) and v(x) are two given function

 

[Office_Shredder]

Try looking at the two cases when u(x)>v(x) and when v(x)>u(x)

 

[AlephZero]

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

 

[micromass]

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.