Proof with partial derivatives

In summary, the conversation is about trying to prove that if a certain statement holds for all (x,y) in the domain, then f must be a linear function. The person has tried different functions and it seems to work, but they are unable to explain why. The other person suggests that if f is linear, it can be represented as Ax+By+C, which may help with the proof.
  • #1
rman144
35
0
I've been trying to prove that if the following statement holds for all (x,y)ER^2, f must be a linear function:

f(x,y)-f(0,0)=x*(d/dx)[f(x,y)]+y*(d/dy)[f(x,y)]

It seems to work for any function I plug in, but I'm unable to establish why this always works. Also, when I say (d/dx)[f(x,y)], I mean the derivative of f(x,y) with respect to x.

Thanks in advance for any help or ideas.
 
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  • #2
IF f(x,y) is a LINEAR function, then you know thay f(x,y)=Ax+By+C, with A, B and C being constants.

Thus, f(x,y)-f(0,0)=Ax+By.

Does that help?
 

1. What is the purpose of using partial derivatives in proofs?

Partial derivatives are used in proofs to determine the rate of change of a function in a specific direction. This is particularly useful in multivariable calculus, where a function can have multiple input variables, and understanding how it changes in each direction is important.

2. How do you find partial derivatives?

To find the partial derivative of a function with respect to a specific variable, you hold all other variables constant and take the derivative as you would for a single-variable function. For example, to find the partial derivative of f(x,y) with respect to x, you would treat y as a constant and take the derivative of f(x,y) with respect to x.

3. Can partial derivatives be used to prove the existence of extrema?

Yes, partial derivatives can be used to prove the existence of extrema (maximum or minimum points) for a function. This is done by setting the partial derivatives equal to zero and solving for the input variables. The resulting points are potential extrema, and further analysis is needed to determine if they are maximum or minimum points.

4. How are partial derivatives related to total derivatives?

Partial derivatives are a special case of total derivatives, which take into account all variables in a function. Partial derivatives only consider the rate of change in one direction, while total derivatives consider the overall rate of change of a function in all directions.

5. Can partial derivatives be used to calculate higher-order derivatives?

Yes, partial derivatives can be used to calculate higher-order derivatives of multivariable functions. This is done by taking multiple partial derivatives of the function, with each subsequent derivative representing a higher order of differentiation.

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