Differentation of Partial Derivative with respective to high order

In summary, Differentiation of Partial Derivative with respective to high order is a difficult concept to understand and visualize. Without help, it is not possible to solve the problem.
  • #1
danong
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[SOLVED, THANKS]Differentation of Partial Derivative with respective to high order

hi there, i am actually studying about functional equation.
I got stucked with some derivatives problem,
and where i could find nowhere to refer or study from,
because it seems it is out of university book level.

my question is this :

what does it means by taking derivative with respect to partial derivative?
can anyone visualize this idea to me?
because i couldn't figure out the term with respect to partial derivative,
when it comes to a functional equation,
of which is a differential equations.

For e.g : F(x,y(x),y'(x),y''(x)) , find [tex]\frac{d}{dx}[/tex] of [tex]\partial[/tex] F(x,y(x),y'(x),y''(x)) / [tex] \partial [/tex] y' .

How can we write the full solution with partial derivative respect to y' ? and how bout y''?
 
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  • #2
Use the chain rule.
 
  • #3
for example?
 
  • #4
How would you usually apply the chain rule for a function with multiple variables?
 
  • #5
sorry i couldn't understand, can you show me the way?
much appreciated.
 
  • #6
You couldn't understand what exactly? Are you telling me that you don't know what the chain rule is?
 
  • #7
yeah, sort of, I'm not good in this, can u show me the general solution with respect to this particular problem?

Please ... if you know how, just show me the solution, don't try to test my skills, i am not good in this. that's why i need help.

Thanks in advance.

God Bless your day.
 
  • #9
okay thanks for sharing,
now i fully understand the rule of this chain rule,
but still don't quite sure how to solve the problem stated above.

could you please provide the solution?


Thanks in advance.

God bless~
 
  • #10
If you fully understand the chain rule now you should be able to make an attempt. Show us such an attempt so we can see where you get stuck.
 
  • #11
God bless, i solved it.
 
  • #12
Good job, Cyosis.
 
  • #13
yah ^^ he did a good job *thumbs up* =)
thanks for everything~
 
  • #14
This forum really need a lot of people like Cyosis, so that everyone who post questions eventually choose to answer themselves, cheers~ ^^
 

Related to Differentation of Partial Derivative with respective to high order

1. What is the definition of differentiation of partial derivative with respect to high order?

The differentiation of partial derivatives with respect to high order is the process of finding the rate of change of a function with respect to multiple independent variables, where each variable is differentiated multiple times. This allows us to understand how small changes in one variable affect the overall behavior of the function.

2. How is the notation for high order partial derivatives written?

The notation for high order partial derivatives is written as follows: f(n)xy, where n represents the order of differentiation and x and y represent the independent variables. This notation is also known as the "nth order mixed partial derivative."

3. What is the difference between high order partial derivatives and total derivatives?

The main difference between high order partial derivatives and total derivatives is the number of independent variables involved. High order partial derivatives involve multiple independent variables, while total derivatives involve only one independent variable.

4. How is the chain rule applied to high order partial derivatives?

The chain rule is applied to high order partial derivatives by taking the derivative of each independent variable separately and multiplying them together. For example, if we have a function f(x,y) = x2y3, then the second order partial derivative with respect to x would be f(2)xx = 2y3, and the second order partial derivative with respect to y would be f(2)yy = 6x2y.

5. What are the applications of high order partial derivatives in science and engineering?

High order partial derivatives have many applications in science and engineering, such as in thermodynamics, fluid dynamics, and mechanics. They are also used in optimization problems to find the maximum or minimum values of a function. In addition, they play a crucial role in understanding and modeling complex systems that involve multiple variables and their interactions.

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