Can the 2nd Order Partial Derivative of a Function be Evaluated at (0,0)?

In summary, the conversation is about evaluating the second order partial derivative of a function that is defined as 0 at (0,0) and has a complicated expression for all other points. The question is whether the derivative at (0,0) is simply 0 or undefined. It is mentioned that the derivative is defined for all points except (0,0) and that its derivative at those points can be found using the product rule. It is also noted that if the function is differentiable, its derivative is continuous and can be found at (0,0) by taking the limit. However, if the function is not differentiable, the derivative at (0,0) is undefined and a rigorous approach using the definition of
  • #1
precondition
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This is supposed to be year1 calculus question but I can't answer it.
If f:R_2-->R is 0 if (x,y)=(0,0) and xy(x_2-y_2)/(x_2+y_2) otherwise then evaluate 2nd order partial derivative DxDyf(0,0) and Dy,Dxf(0,0)
The thing is, I get some complicated looking expression for DxDyf(x,y) and I can't simply put x=0 and y=0 in that expression right? because it gives 0/0... What am I misunderstanding here?
 
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  • #2
or is it that DxDyf(0,0) is simply zero because at (0,0) the function is 0?
 
  • #3
The derivative is defined for all (x,y) except (0,0), and its derivative at these points is just what you probably found (using the product rule, etc). If the function is differentiable, then its derivative is continuous, and so you can find the derivative at (0,0) by taking the limit. If not, then the derivative at (0,0) is undefined. If you need to be rigorous, you'll need to go back to the definition of the derivative in terms of limits.
 

Related to Can the 2nd Order Partial Derivative of a Function be Evaluated at (0,0)?

1. What is "Easy partial differentiation"?

Easy partial differentiation is a mathematical technique used in calculus to find the rate of change of a function with respect to one of its variables while holding all other variables constant. It is useful for solving problems involving multiple variables, such as in physics, economics, and engineering.

2. How is "Easy partial differentiation" different from regular differentiation?

Regular differentiation, also known as ordinary differentiation, is used to find the derivative of a function with respect to a single variable. Easy partial differentiation, on the other hand, finds the partial derivative of a function with respect to one of its variables while treating all other variables as constants.

3. What is the notation used for "Easy partial differentiation"?

The partial derivative of a function f with respect to a variable x is denoted by ∂f/∂x or fx. It is read as "the partial derivative of f with respect to x" or "f sub x."

4. What are some common applications of "Easy partial differentiation"?

Easy partial differentiation has many applications in science and engineering. It is used to find the maximum or minimum values of a function, to determine the direction of steepest ascent or descent of a function, and to solve optimization problems in various fields such as economics, physics, and chemistry. It is also used to find the rate of change of a function in a specific direction, known as directional derivatives.

5. What are some tips for solving problems involving "Easy partial differentiation"?

Some tips for solving problems involving easy partial differentiation include understanding the concept of partial derivatives, knowing the rules for finding partial derivatives, and practicing with different types of functions. It is also important to carefully identify which variables are being held constant and which ones are being differentiated with respect to. Additionally, using software programs such as Mathematica or Wolfram Alpha can help in solving more complex problems involving partial differentiation.

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