Understanding Absolute Value Sign in Derivative

In summary, the conversation discusses the confusion about the absolute value sign in a given derivative and the process of setting it to zero. It is clarified that the expression represents a conditional function with two separate functions for x greater than or less than 1, but not at x=1. It is suggested to plot the function to better understand it.
  • #1
sony
104
0
Hi

So I am a little confused about the absolute value sign here. Is the derivative right:

[x-1/|x-1|]e^x + |x-1|e^x

But what do I do when I set this to zero? Do I get two expressions, one for x>1 and one for x<1 ? (thus removing the absolute value signs)
 
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  • #2
Hints anyone...?
 
  • #3
[tex]|x-1|=x-1\quad\text{for}\quad x\geq 1[/tex]

and:

[tex]|x-1|=-(x-1)\quad\text{for}\quad x<1[/tex]

Thus your expression represents this function:

[tex]y(x)=
\left\{
\begin{array}{rcl}
(x-1)e^x & \mbox{for} & x\geq 1 \\
-(x-1)e^x & \mbox{for} & x<1
\end{array}\right.
[/tex]

and so the derivative will represent two functions likewise but will not exist at x=1. Try plotting this conditional function and you'll see what I mean.
 

1. What is the absolute value sign in a derivative?

The absolute value sign in a derivative represents the distance between a point on a graph and the x-axis. It can also be thought of as the magnitude or size of a number, without considering its direction.

2. Why is the absolute value sign used in derivatives?

The absolute value sign is used in derivatives because it allows for the calculation of the slope of a line at any given point, regardless of whether the slope is positive or negative. It also helps to simplify the derivative equation and make it easier to work with.

3. How is the absolute value sign used to find the derivative of a function?

The absolute value sign is used by taking the derivative of the function as normal, and then taking the absolute value of the resulting expression. This allows for the calculation of the slope at any given point on the function.

4. Can the absolute value sign be used in all types of derivatives?

Yes, the absolute value sign can be used in all types of derivatives, including single-variable, multivariable, and partial derivatives. It is a universal mathematical concept that is applicable in all types of derivatives.

5. What is the significance of the absolute value sign in finding critical points?

The absolute value sign is important in finding critical points because it helps to identify the points where the derivative is equal to zero, regardless of whether the slope is increasing or decreasing at that point. This allows for the identification of key points on a function that may have important implications in real-world applications.

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