I would think that if you can find the derivative using calculus, then the derivative has to exist, and you would not need a computer.

If you cannot calculate the derivative, then you might use a computer to see how the function behaves as the independent variable is changed, or watch how the function blows up? I'm assuming you are talking about functions with singularities?

Conceivably, using this cheating method, you might miss some singularities, depending how you change the independent variable. You might just skip over a singularity?

Thanks Albert M. I am working in the automation of a process in that i require partial derivatives. I use numerical differentiation.The derivative does not exist at x = 0 for | x | , but using numerical differentiation makes an error if you apply formula. How can you include additional information in calculating the numerical derivative without being specific in each of the functions?

Thanks Peter for your answer. I thought that there was some form to know if the function is derivative in a specific point using only computational calculus. 

Thanks.