Derivative Of Absolute Value Function

Derivative Of Absolute Value Function

Derivative Of Absolute Value: In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, and |x| = −x for a negative x (in which case x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

Derivative Of Absolute Value Function

Derivative Of Absolute Value

In the video lecture that I am watching, the instructor compared an example of finding the derivative of the f(x)=|x|f(x)=|x| vs f(x)=x|x|f(x)=x|x|, where f(x)=x|x|f(x)=x|x|, derivative is 00, whereas f(x)=|x|f(x)=|x|, does not have a derivative.

The instructor highlighted that the absolute value function does not have a derivative compared to f(x)=x|x|f(x)=x|x|.

If I would to apply the power rule: f(x)=nxn1f(x)=nxn−1, where f(0)=1011f(0)=1∗01−1. Because 0 to the power of 0 is undefined.

Is it because 00 to the power of 00 is undefined, therefore the absolute value function does not has a derivative?

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The power rule only applies to power functions. Neither x|x|x↦|x| nor xx|x|x↦x|x| are power functions. So it isn’t relevant here—you have to go back to the definition to decide if the function has a derivative. Possible duplicate of math. Matthew Leingang, in this case, the exponent of X must not be equal to zero? x^2 or x^-2 should be a valid power function?
Yes, x2x2 and x2x−2 are valid power functions. But your question was about |x||x| and x|x|x|x|, right? Those are not power functions. You could insert exponents, but it would have to be 11, not 00. That doesn’t help with your calculation, though. Applying the power rule to |x|1|x|1 gives 11, times |x|0|x|0 (also 11), times the derivative of |x||x|.

Derivative Of An Absolute Value

Derivative Of An Absolute Value

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 number, 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 are a sign and a value. We could write +3 as:

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

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.

How To Take Derivative Of Absolute Value

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

Is absolute value differentiable?

When a function is differentiable it is also continuous. But a function can be continuous but not differentiable. For example, the absolute value function is actually continuous (though not differentiable) at x=0.

Is absolute value continuous?

The absolute value function is continuous at 0. The absolute value function is not differentiable at 0.

What is the definition of a derivative?

In mathematics, the derivative is a way to show the rate of change: that is, the amount by which a function is changing at one given point. For functions that act on the real numbers, it is the slope of the tangent line at a point on a graph.