# Derivative of min(.)

1. Jun 22, 2005

I'm trying to take the derivative of a min function.

I have some function that depends on the variable x and a set of parameters x_i = x_1, x_2, ... .

$$f_i(x) = g(x,x_i)$$

and then

$$y = min_i(f_i(x))$$

So I'm finding the minimum value of f over all the x_i for some particular x value. Now I want to take dy/dx.

Is there some definition of min that allows differentiation? Like maybe calling it the $$\frac{1}{\infty}$$ norm. Although I don't think that would help since I probably can't differentiate infinite exponents. I don't think d/dx can pass through the min because that would just give me the minimum derivative value corresponding to one of the x_i, but what I want is how the minimum over all i changes as x moves relative to the x_i.

Any help?

2. Jun 22, 2005

### lurflurf

Remeber min(a,b)=.5(a+b-|a-b|)
In other words it is easy to differentiate min, but the derivative will not exist at points where two values are equal but the derivatives are not. This is clear as min often has corners. Also of interest is min(a(i))=-max(-a(i)).
Example y=min(0,x)=x x<=0, 0 x>=0
y'=1 x<0, 0 x>0 (undefined for x=0)

3. Jun 23, 2005