# 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