Taking second derivative of a derivative

  • Thread starter nkinar
  • Start date
75
0
Hello--

I'm in the process of implementing a PML for FDTD modeling.

I would like to take the derivative of the partial derivative shown below, but I am uncertain with respect to how I might proceed.

[tex]
\[
\frac{\partial }{{\partial x}} \to \frac{1}{{1 + \frac{{i\sigma \left( x \right)}}{\omega }}}\frac{\partial }{{\partial x}}
\]
[/tex]

Essentially what I would like to do is take the derivative of a partial derivative, and also deal with the [tex]\[{i\sigma \left( x \right)}\] [/tex] term, which is a function of position [tex]x[/tex].

This would result in the calculation of [tex] \[\frac{{\partial ^2 }}{{\partial x^2 }}\][/tex]
 
Last edited:
75
0
Perhaps this would be the way to take the second derivative :

[tex]
\[
\frac{{\partial ^2 }}{{\partial x^2 }} \to -\left( {1 + \frac{{i\sigma \left( x \right)}}{\omega }} \right)^{ - 2} \left( {\frac{{\partial \sigma \left( x \right)}}{{\partial x}}\frac{i}{\omega }} \right)\frac{\partial }{{\partial x}} + \frac{{\partial ^2 }}{{\partial x^2 }}\left( {1 + \frac{{i\sigma \left( x \right)}}{\omega }} \right)^{ - 1}
\]

[/tex]
 
Last edited:

HallsofIvy

Science Advisor
Homework Helper
41,698
871
Use the product rule:
[tex]\frac{\partial }{\partial x}\left((1+ \frac{i\sigma}{\omega})^{-1}[/tex][tex]\frac{\partial y}{\partial x}\right)[/tex][tex]= \frac{\partial (1+ \frac{i\sigma}{\omega})^{-1}}{\partial y}{\partial x}[/tex][tex]+ (1+ \frac{i\sigma}{\omega})^{-1}\frac{\partial^2 y}{\partial x^2}[/tex]
 
Last edited by a moderator:

Related Threads for: Taking second derivative of a derivative

Replies
6
Views
2K
  • Posted
Replies
4
Views
2K
Replies
2
Views
3K
  • Posted
Replies
5
Views
2K
  • Posted
Replies
11
Views
2K
Replies
1
Views
2K
  • Posted
Replies
5
Views
1K
  • Posted
Replies
1
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top