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Given the function:

$\ f(x,y,z) = x^2 + y^2 + \arctan(z) $

How can we determine if the differential $[ \mathrm{d}f(1,2,3) ]$ exists and then calculate it?

Finding all the partial derivatives and confirming that they are continuous is enough? (because we have a theorem that says "if the partial derivatives of $f$ at $(x_0, y_0, z_0)$ are continuous then there is the differential of $f$ at the specific point".

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If the partial derivatives of $f$ exist and are continuous in a open neighborhood of $(x_0,y_0,z_0)$, then the differential of $f$ exist at $(x_0,y_0,z_0)$. You can see that for $f$, his partial derivatives exist and are continuous in very big neighborhood of $(1,2,3)$ –  leo Apr 21 '12 at 3:53
    
@leo: Thank you leo! You could change your comment to an anwser! I just wanted to be sure :) –  Chris Apr 21 '12 at 15:30

1 Answer 1

Moving leo's comment into answer:

If the partial derivatives of $f$ exist and are continuous in a open neighborhood of $(x_0,y_0,z_0)$, then the differential of $f$ exist at $(x_0,y_0,z_0)$. You can see that for $f$, his partial derivatives exist and are continuous in very big neighborhood of $(1,2,3)$ – leo Apr 21 at 3:53

[Indeed they are continuous on all of $\mathbb R^3$]

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