Is f(x,y) Differentiable at (0,0)?

In summary, the function f(x,y)=\frac{x^3}{x^2+y^2} is not differentiable at (0,0) since the partial derivative with respect to y is not continuous at that point. This is shown by taking the limit when approaching (0,0) in different directions. Therefore, f(x,y) has no derivative at (0,0).
  • #1
Barioth
49
0
Hi,

I'm having trouble évaluation the differentiability at (0,0) of the function

\(\displaystyle f(x,y)=\frac{x^3}{x^2+y^2}\) for (x,y) not nul, and \(\displaystyle f(x,y)=0\) if (x,y)=0

I know f is differentiable if (x,y) isn't nul since the partial derivative are continuous, but I don't know how to evaluate it at (0,0)

Thanks for passing by
 
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  • #2
Barioth said:
Hi,

I'm having trouble évaluation the differentiability at (0,0) of the function

\(\displaystyle f(x,y)=\frac{x^3}{x^2+y^2}\) for (x,y) not nul, and \(\displaystyle f(x,y)=0\) if (x,y)=0

I know f is differentiable if (x,y) isn't nul since the partial derivative are continuous, but I don't know how to evaluate it at (0,0)

Thanks for passing by

Let's take the partial derivative with respect to y:
$$\frac{\partial}{\partial y}f(x,y) = - \frac{2x^3 y}{(x^2+y^2)^2}$$

Is it the same in all directions when approaching (0,0)?
You may want to switch to polar coordinates.
 
  • #3
I like Serena said:
$$\frac{\partial}{\partial y}f(x,y) = - \frac{2x^3 y}{(x^2+y^2)^2}$$

tacking y=m*x we end up with

\(\displaystyle \frac{-2*m*x^4}{(m^2+1)^2*x^4}= \frac{-2*m}{(m²+1)²}\)

This telling me that the limite when (x,y)->(0,0) isn't contiunous.

There for the partial derivative doesn't exist at 0. I can conclude that f(x,y) has no derivative at (0,0).

Thank you for the help!
 

FAQ: Is f(x,y) Differentiable at (0,0)?

What is differentiability at (0,0)?

Differentiability at (0,0) refers to the mathematical concept of a function being smooth and having a defined slope at the point (0,0) on a graph.

How is differentiability at (0,0) determined?

Differentiability at (0,0) is determined by taking the limit of the average rate of change of a function as the interval approaches 0. If the limit exists, the function is differentiable at (0,0).

Why is differentiability at (0,0) important?

Differentiability at (0,0) is important because it indicates that a function is continuous and has a defined slope at a specific point, which allows for the use of calculus to analyze and solve problems involving the function.

Can a function be continuous but not differentiable at (0,0)?

Yes, a function can be continuous but not differentiable at (0,0). For a function to be continuous, it must have a defined value at (0,0) and the limit of the function as x approaches 0 must also exist. However, the function may still not have a defined slope at (0,0), meaning it is not differentiable.

What are some real-life examples of differentiability at (0,0)?

A common real-life example of differentiability at (0,0) is the velocity of an object at a specific point in time. The velocity is the slope of the object's position-time graph, and at any given moment, the velocity is the average rate of change of the object's position as the time interval approaches 0. Other examples include the temperature change at a specific point, the rate of change of a stock price at a specific time, and the acceleration of a moving object at a particular point.

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