Calculating Derivatives of Absolute Value Functions

Calculating Derivatives of Absolute Value Functions
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  • 0:00 Calculating…
  • 0:31 The Signum Function
  • 2:34 The Absolute Value Function
  • 3:39 Derivative of f(x) = |x|
  • 5:18 Find the Derivative of…
  • 5:57 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

A fundamental distance measure in math, science, and engineering is the absolute value. In this lesson, we will explore the differentiation of absolute value functions.

Calculating Derivatives of Absolute Value Functions

Valerie is trekking through the mountainous region behind the shopping mall. Her map shows the nearest coffee place is either two miles north or minus three miles south. Ignoring plus and minus signs, Valerie considers only absolute values.

Absolute value functions are common in math. In this lesson, we will show how to differentiate these absolute value functions, but first we will discuss the signum function. This gives Val more time to plan a route.

The Signum Function

Imagine Val has a special hiking bag. Inside this bag are numbers, like 3, -4, 6, and -17. We just happened to pull out some integers, but the bag contains fractions and decimal numbers, as well. All numbers have two parts: a sign and a value. These four numbers are actually +3, -4, +6, and -17. There's a sign and a value. We could write +3 as:


+3_as_sgn(+3)_abs(+3)


and -17 as:


-17_as_sgn(-17)_abs(-17)


The vertical lines mean 'take the absolute value of the number enclosed.' For example, |+3| is 3 and |-17| is 17. sgn( ) is the signum function. If the number is positive, sgn is positive. If the number is negative, sgn is negative. If the number is 0, sgn is 0. For the variable x, sgn(x):


signum


Instead of a specific number like +3, think of the variable x. Comparing to +3 = sgn(+3) |+3|, we write x = sgn(x) |x|. Solve x = sgn(x) |x| for sgn(x) by dividing both sides of the equation by |x|. This gives an alternative definition for sgn(x):


signum_definition


Note: To avoid dividing by zero, this alternate definition of signum is good for all values of x except x = 0.

The Absolute Value Function

For f(x) = |x|, f(x) is positive for both negative and positive x. If x is 0, f(x) = 0. As a plot:


The absolute value function
f(x)=|x|


See the negative slope when x is negative? There's a positive slope when x is positive. At x = 0, the sharp turning point means the slope is undefined. This sounds like the signum function! In fact, we are ready to show that the signum is the derivative of the absolute value function.

The square root of 4 is ± 2. We have two possible signs with the square root. To relate the derivative of the absolute value to the signum, express the absolute value of x as the unsigned square root of x squared:


absolute_value_as_square_root_of_x_squared


Val has decided on the coffee shop located plus two miles north. Two reasons for the choice: first, in the absolute value of the distances, two miles is the shortest. Second, she likes the name of the shop: 'Derivative Espressos and More.'

Derivative of f(x) = |x|

Let's show how f '(x) = sgn(x) for f(x) = |x|.

Start by writing |x| as the unsigned square root of x2:


unsigned_square_root


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