Partial Derivative of f(x,y) at (0,0): Find & Evaluate

In summary, the conversation is about finding the partial derivative of a function with respect to x and evaluating it at a specific point. The attempted solution involves using the limit definition of a derivative, and the final answer depends on how the point is approached. However, the question itself is flawed as the partial derivative is not defined at the given point.
  • #1
zhuyilun
27
0

Homework Statement



find the partial derivative of f(x,y)=(x^3+y^3)^(1/3) with respect to x and evaluate at (0,0)

Homework Equations





The Attempt at a Solution


i found the general partial derivative with respect to x is (x^2)*(x^3+y^3)^(-2/3)
if i plug in the point i would get zero at the bottom
so i used the limit thing which is the limit of (f‘(x+h,y)-f(x,y))/h as h approaches infinite.
then i substitute , i got something like lim (((x+h)^3+y^3)^(1/3)-(x^3+y^3)^(1/3))/h as h approaches infinite. then i plug in x=0, y=0, i got lim ((h^3)^(1/3))/h as h approaches infinite which is just 1
i am not sure about what i did is right or not
 
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  • #2
Your partial derivative is correct, but the value at (x,y)=(0,0) depends on how you approach this point. It is 1 if you first set y=0 with x positive, and then take the limit as x->0.

Bad question. Complain to your instructor. Seriously.
 
  • #3
The partial derivative is simply not defined at (0,0).
 

1. What is a partial derivative?

A partial derivative is a mathematical concept used to measure the rate of change of a function with respect to one of its variables, while holding the other variables constant.

2. How is the partial derivative of a function at a specific point evaluated?

To evaluate the partial derivative of a function at a specific point, we use the limit definition of a derivative. We take the limit as the variable approaches the given point, while keeping other variables constant.

3. What is the notation used for partial derivatives?

The notation used for partial derivatives is similar to that of regular derivatives, except we use subscripts to indicate which variable we are taking the derivative with respect to. For example, ∂f/∂x represents the partial derivative of a function f with respect to x.

4. How do we interpret the value of a partial derivative?

The value of a partial derivative at a specific point represents the slope of the tangent line to the function at that point, in the direction of the variable for which the derivative is taken. It tells us how much the function is changing in that specific direction.

5. Why is the partial derivative at (0,0) important?

The partial derivative at (0,0) is important because it represents the rate of change of the function at the origin, which is a critical point. This value can provide insight into the behavior of the function and its critical points.

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