Derivatives of Absolute Value Functions

Tutorial on how to find derivatives of functions in calculus (Differentiation) involving absolute value functions.

Example 1: Find the first derivative f '(x), if f is given by

f(x) = |x - 1|

Solution to Example 1
Noting that | u | = √ ( u2 ), let u = x - 1 so that f(x) may be written as
f(x) = y = √( u2 )
We now use the chain rule
  f '(x) = (dy / du) (du / dx)
= (1/2) (2 u) / √ (u2) (du / dx)
= u . u' / | u |
= u . 1 / √ (u2) = (x - 1) / |x - 1|
We note that if x > 1, |x - 1| = x - 1 and f '(x) = 1. If x < 1, |x - 1| = -(x - 1) and f '(x) = -1. f'(x) does not exist at x = 1.

Example 2: Find the first derivative of f given by

f(x) = - x + 2 + |- x + 2|

Solution to Example 2
f(x) is made up of the sum of two functions. Let u = - x + 2 so that
f'(x) = -1 + u u' / |u| = -1 + (- x + 2)(-1) / |-x + 2|
We note that if x < 2, |- x + 2 | = - x + 2 and f'(x) = -2 . If x > 2, f'(x) = 0. f'(x) does not exist at x = 2.
As an exercise, plot the graph of f and explain the results concerning f'(x) obtained above.

Exercises: Find the first derivatives of these functions
1. f(x) = |2x - 5|
2. f(x) = (x - 2)2 + |x - 2|


Answers to above exercises:
1. f'(x) = 2 (2x - 5) / |2x - 5|
2. f(x) = 2 (x - 2) + (x - 2) / |x - 2|


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differentiation and derivatives