# Math Help - Finding a "best" derivative of a function that is not differentiable at a point

1. ## Finding a "best" derivative of a function that is not differentiable at a point

Hi everyone,
I have a topic I am suppose to investigate and write a small paper about. The question deals with functions that are non-differentiable at one or more points, but "looks like" it "could" have a horizontal tangent at one or more of these points if it could. For example, y = |x| is clearly not differentiable at x = 0 and thus has no horizontal tangent line, but the most natural tangent line there seems to be at x=0, which reveals a relative minima at (0,0). So, the question is what do you do/set equal to zero with these kinds of functions i.e. is there some "derivative-like" quantity/formula analogous to the regular definition of the derivative that would produce these values (x values that appear to have a "best" tangent line despite not being differentiable there). I would then need go into detail about generalzing all the regular theorems such as product, quotient, chain, IVT, etc., but first I need some basic direction on where to start.

james

2. ## Re: Finding a "best" derivative of a function that is not differentiable at a point

Hey james121515.

In short, are you trying to find derivatives or maxima/minima for functions where derivatives don't exist?

One suggestion I have is if you fit the function to one that is continuous and differentiable by projecting it to say an orthogonal fourier polynomial or orthogonal function basis.