Question

What is the derivative of f(x)=l x l at x=0?
a) 1
b) -1
c) 0
d) Does not exist.

Answer 1

then ofcourse it should be -1, right?

Answer 2

not sure,.....if f'(x) = -1, if x < 0;
f'(x) = 1, if x > 0

Answer 3

from another angle....x-->p- means from the left (or negative side).
x-->p+ means from the right (or positive side).

But

limit |x| = limit of -x = -1 at p (the slope of the line) .
x-->0-

limit |x| limit of x = +1 at p (the slope of the line) .
x-->0+

Since there is no unique limit at 0 for f(x) = |x|, then f(x) = |x| has no derivative at 0

Answer 4

thanks for your valuable tips.....