Find ∂/∂y ∂F/dx (0,0) given F(x,y)

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In summary, the conversation was about taking derivatives of a function and determining its continuity at (0,0). The attempt at a solution involved finding the limit of the differentiated function, but it was found to not exist. It was then realized that the function may not be continuous, leading to confusion on how to proceed. Two different second derivatives were tried, showing that they give different results.
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s3a
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Homework Statement


Attached.


Homework Equations


Just taking derivatives.


The Attempt at a Solution


Basically, at first I was thinking I can differentiate because the original function is continuous at (0,0) so I do and get: http://www.wolframalpha.com/input/?i=d/dy+(d/dx+(xy(8x^2+%2B+4y^2)/(x^2%2By^2))). Correct me if I am wrong but I was thinking I had to differentiate this twice partially as the Wolfram Alpha link shows and then plug in (0,0) to the answer that Wolfram Alpha gave but that is also not defined. I tried taking the limit with y = 0 and x = 0 respectively and found that the limit as (x,y) => (0,0) of the differentiated part does not exist. I then thought that wait, having F(0,0) = 0 be defined does not mean that it is continuous so I shouldn't be able to differentiate in the first place. I am now completely confused as to what I need to do. :cry:
 

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  • #2
Try
d/dx (x*y*(8*x^2+4*y^2)/(x^2+y^2)) for x=0
and you get 4y
Next try
d/dy (x*y*(8*x^2+4*y^2)/(x^2+y^2)) for y=0
and you get 8x
So the second derivatives will give you different results.
 

1. What does "Find ∂/∂y ∂F/dx (0,0) given F(x,y)" mean?

This notation represents the partial derivative of the function F with respect to x, evaluated at the point (0,0). It is asking for the rate of change of F in the y direction, while holding x constant, at the point (0,0).

2. How do I find the partial derivative of a function?

To find the partial derivative of a function, you need to differentiate the function with respect to the specific variable while holding all other variables constant. This means treating all other variables as constants and using the basic rules of differentiation.

3. Why is it important to evaluate the partial derivative at a specific point?

Evaluating the partial derivative at a specific point allows us to determine the instantaneous rate of change of the function at that point. This can be useful in understanding the behavior of a function and making predictions about its behavior in the surrounding points.

4. How do I evaluate a partial derivative at a specific point?

To evaluate a partial derivative at a specific point, you can plug in the given values for the variables into the derivative expression and solve for the result. In this case, you would plug in x=0 and y=0 into the expression for ∂F/dx and then plug that result into the expression for ∂/∂y.

5. What are some real-world applications of partial derivatives?

Partial derivatives have many real-world applications, including in economics, physics, and engineering. They can be used to determine optimal production levels, analyze the behavior of complex systems, and optimize functions in various fields.

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