Calculating errors in Functions of two variables Taylor Series

[thomas49th]

Homework Statement

From the taylor series we can replace x =x_{0} + h
but how does
\delta f = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0})
become
\delta f = hf(x_{0}, y_{0}) + kf(x_{0}, y_{0})
I can see the first step, but how do you get it to the second?

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

From the taylor series we can replace x =x_{0} + h
but how does
\delta f = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0})
become
\delta f = hf(x_{0}, y_{0}) + kf(x_{0}, y_{0})
I can see the first step, but how do you get it to the second?

Homework Equations

The Attempt at a Solution

\Delta f(x, y) = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0})
\approx f(x_{0}, y_{0}) + f_x(x_0, y_0)\Delta x + f_y(x_0, y_0)\Delta y - f(x_{0},y_{0})

In the Taylor expansion above, terms of order 2 and higher are omitted.