This equality holds true for every value. Can you prove it?

Question

This equality holds true for every value. Can you prove this identity?
d/dx |x| = |x|/x = x/|x|

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Answer 1

If x > 0, then the derivative is 1.......... .If x < 0, then the derivative is -1. Since there is no derivative for |x| at x = 0, then it seems that that is 0/0........

^_^ (h3h3h3!!)

Answer 2

If x is positive, then |x| = x. So d/dx( |x| ) = d/dx( x ) = 1 = x / x = |x| / x = x / |x|.

If x is negative, then |x| = -x. So d/dx( |x| ) = d/dx ( -x) = -1 = -x / x = x / -x = |x| / x = x / |x|.

The equality fails for x = 0 since then both |x| / x and x / |x| would be 0 / 0, which is undefined.

Answer 3

This equality holds true for every value. Can you prove this identity?
d/dx |x| = |x|/x = x/|x|

It is NOT valid for x=0

if x>0: |x| =x so, d/dx |x| =d/dx x =1
and |x|/x=x/|x| = x/x =1

if x<0: |x| = -x so, d/dx |x| =d/dx -x = -1
and |x|/x=-x/x = x/-x =x/|x| = -1 .