How To Determine The Differentiability Of A Function F(X,Y) In Mathematics.

Differentiability of f(x,y)

if the partial derivatives fx and fy exist near (a,b) and are continuous at (a,b), then f(x,y) is differentiable at (a,b)

To determine the differentiability of a function f(x,y), we need to check if there exist partial derivatives of f(x,y) with respect to x and y, and whether these partial derivatives are continuous at the point of interest.

If the partial derivatives exist and are continuous at the point (a,b), then the function f(x,y) is said to be differentiable at (a,b). If the partial derivatives do not exist or are not continuous at (a,b), then the function f(x,y) is said to be non-differentiable at (a,b).

This can be mathematically expressed as follows:

1. Find the partial derivatives of f(x,y) with respect to x and y:

fx(x,y) = ∂f/∂x
fy(x,y) = ∂f/∂y

2. Check if fx and fy exist at the point (a,b) by evaluating the limit of the difference quotient:

lim Δx→0 (f(a+Δx,b) – f(a,b))/Δx = fx(a,b)
lim Δy→0 (f(a,b+Δy) – f(a,b))/Δy = fy(a,b)

If the limits exist, then fx and fy exist at (a,b).

3. Check if fx and fy are continuous at (a,b):

If fx and fy are continuous at (a,b), then f(x,y) is differentiable at (a,b). Otherwise, it is non-differentiable at (a,b).

In summary, the differentiability of a function f(x,y) can be determined by finding its partial derivatives with respect to x and y, evaluating these derivatives at the point of interest, and checking their continuity.

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