0
votes
5answers
87 views

Why is $f(x)=|x|$ not differentiable?

Consider the function $f(x)=|x|$, I know that $f$ is not differentiable at $x=0$, but still, when you try to differentiate $f(x)=\sqrt{x^2}$ (which is exactly the same), you get: ...
0
votes
1answer
30 views

How to take derivative of sums of absolute values

Take the derivative of $f(m) = \sum_i | x_i - m |$. I've been told that derivative of each term is +1 or -1. How do you show that?
0
votes
1answer
21 views

Calculate the area that the following graphs form

I have been trying and trying to solve the following problem (I even used wolframalpha as an extra help, but no success, and I have like 100 calculations in my notebook): The Task: Calculate the ...
6
votes
6answers
126 views

why minimum of these functions happen at a special place?

why minimum of these functions happen at a special place? how to use derivative to find the minimum of these functions? $$|x-1| + |x-2| + \dots + |x-9|$$ minimum is for $x = 5$ $$|x-1| + |x-2| + \dots ...
1
vote
1answer
67 views
0
votes
1answer
2k views

Chain rule and the derivative of absolute value functions

Is it possible to use the chain rule to calculate the derivative of $|x^4|$ and $|x|^4$ in $x=0$? Does the derivative to these functions exist in $x=0$?
3
votes
3answers
130 views

Finding the derivative of $|x|^4$ using the chain rule.

I am presented with the following task: Can you use the chain rule to find the derivatives of $|x|^4$ and $|x^4|$ in $x = 0$? Do the derivatives exist in $x = 0$? I solved the task in a rather ...
0
votes
1answer
131 views

For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?

I came across the following question: For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$? What I have tried: Since for $ \alpha = 1 $ is clearly non-differentiable in ...
1
vote
1answer
3k views

Finding the points of the curve where the tangent line is horizontal

The curve given is $\displaystyle y = \ln|x-2| + x + \frac{12}{x-2}$. Find the points of the curve where the tangent line is horizontal. My first stumbling block is the absolute value function. I ...
2
votes
2answers
4k views

How to find critical points of an absolute values function

I am asked to find How many critical points does the function $g(x) = |x^2 − 4|$ have? I know that the result is $3$ but I can only find $2$. What I do, is to equal the equation to $0$, so $x^2-4=0$ ...
2
votes
2answers
217 views

Why is the derivative of $\frac{|x|}{x}$ equal to $\emptyset$ at $x=0$?

I got a bit of a confusion here. If $\varphi(x)=\frac{|x|}{x}$, then $$ \varphi(x) = \left.\Bigg\{ \begin{array}{cc} 1 &if \ x>0\\ \emptyset & if \ x=0\\ -1 & if \ x <0 \end{array} ...
5
votes
1answer
4k views

Derivatives of functions involving absolute value

I noticed that if the absolute value definition $\lvert{x}\rvert=\sqrt{x^2}$ is used then we can get derivatives of functions with absolute value, without having to redefine them as piece-wise. For ...
1
vote
1answer
271 views

The relationship between the derivative of $f(x)$ and $|f(x)|$

I have seen it in an exercise book. I don't know how to do it. If $f(x)$ is continuous at $x=a,$ and $|f(x)|$ can be differentiated at $x=a,$ then $f(x)$ is differentiable at $x=a.$
1
vote
2answers
281 views

Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ...
0
votes
1answer
711 views

derivative of absolute value of a complex function

If $f:U\subset\mathbb{C}\mapsto\mathbb{C}$, where $f(x+iy)=u(x,y)+iv(x,y)$ is a meromorphic function and if $f$, $f'$, and $f''$ are not zero in the strip $a<x<b$, can we get ...