DERIVATIVE OF ABSOLUTE VALUE FUNCTION

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⁄||)(1)

|x|' = ˣ⁄||

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

Let y = |x|'.

Then, we have y = ˣ⁄||.

In y = 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 = -1

when x = -2,

y = -²|| = -²⁄₂ = -1

when x = -1,

y = -¹|| = -¹⁄₁ = -1

when x = 0,

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

when x = 1,

y = ¹|| = ¹⁄₁ = 1

when x = 2,

y = ²|| = ²⁄₂ = 1

when x = 3,

y = 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|.

Find the derivative of each of the following absolute value functions.

Example 1 :

|2x + 1|

Solution :

Example 2 :

|x+  1|

Solution :

Example 3 :

|x|3

Solution :

In the given function |x|3, using chain rule, first we have to find derivative for the exponent 3 and then for |x|.

Example 4 :

|x2 - 5x + 6|

Solution :

Example 5 :

|2x - 5|

Solution :

Example 6 :

(x - 2)2 + |x - 2|

Solution :

Example 7 :

3|5x + 7|

Solution :

Example 8 :

|sinx|

Solution :

Example 9 :

|cosx|

Solution :

Example 10 :

|tanx|

Solution :

Example 11 :

|sinx + cosx|

Solution :

Example 12 :

|cscx|

Solution :

Example 13 :

|secx|

Solution :

Example 14 :

|cotx|

Solution :

Example 15 :

|lnx|

Solution :

Example 16 :

|ex|

Solution :

Example 17 :

|√x|

Solution :

Example 18 :

Solution :

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