Let |f(x)| be an absolute value function.

Then the formula to find the derivative of |f(x)| is given below.

Based on the formula given, let us find the derivative of |x|.

|x|' = (ˣ⁄|ₓ|)(x)'

|x|' = (ˣ⁄|ₓ|)(1)

|x|' = ˣ⁄|ₓ|

Therefore, the derivative of |x| is ˣ⁄|ₓ|. 

Let y = |x|'.

Then, we have y = ˣ⁄|ₓ|.

In y = ˣ⁄|ₓ|, if we substitute x = 0, the denominator becomes zero.

Since the denominator becomes zero, y becomes undefined at x = 0

Let us substitute some random values for x in y. 

when x = -3,

y = -³⁄|₃| = -³⁄₃ = -1

when x = -2,

y = -²⁄|₂| = -²⁄₂ = -1

when x = -1,

y = -¹⁄|₁| = -¹⁄₁ = -1

when x = 0,

y = 0/|0| = 0/0 = undefined

when x = 1,

y = ¹⁄|₁| = ¹⁄₁ = 1

when x = 2,

y = ²⁄|₂| = ²⁄₂ = 1

when x = 3,

y = ³⁄|₃| = ³⁄₃ = 1

Let us summarize the above calculation in table.

Now, based on the table given above, we can get the graph of derivative of |x|.

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