Total differential for a function with two variables

Total differential for a function of two variables is known as

$df = f_x(x_0, y_0)dx + f_y(x_0,y_0)dy$

It's not clear how to derive this. Thomas' Calculus 11th Ed pg 1021 says to use the definition of linearization L(x,y). The definition of linearization for a function f(x) at page 223 is only for one variable

$f(x) \approx L(x) = f(x_0) + f'(x_0)(x-a)$

or

$L(x) = f(x_0) + f'(x_0)∆x$

I had the idea to rewrite the formula $L(x)$ for $L(x,y)$, so I had

$L(x,y) = f(x_0,y) + f_x(x_0,y)∆x$

Then I thought I could maybe linearize $f(x_0,y)$ to get

$L(x,y) = L(x_0,y_0) + f_y(x_0,y_0)∆y + f_x(x_0,y_0)∆x$

So

$∆L = L(x,y) - L(x_0,y_0) = f_y(x_0,y_0)∆y + f_x(x_0,y_0)∆x$

Would this be right?

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Looks good to me. – Grumpy Parsnip Feb 9 '11 at 23:23

Sorry, the "comments" section is disabled, so I will post in here. My apologies if this is against the protocol.

Yes; the total differential is what you just described: the equation of the plane ( as a 2-d linear object) tangent to your function at a point; this tangent plane is the linearization of f, or, in a precise del-eps. sense, the best linear approximation to f, in a small 'hood (neighborhood) of a point.

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It seems the derivation is correct, thanks Jim Conant.

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Definition of total differential:- "If we move form $(a,b)$ to $(a+\mathrm{d}x,b+\mathrm{d}y)$ near by then the result is change."

$df = [fx]\mathrm{d}x + [fy]\mathrm{d}y$.

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