Hence if $a$ varies by $\Delta a$, and if $b$ is held fixed, $$\Delta x \approx \partial_a x \cdot \Delta a.$$
Part b
Similarly to part (a), $$\Delta y \approx \partial_b y \cdot \Delta b.$$
Part c
Similarly to part (a), $$\Delta f \approx \partial_x f \cdot \Delta x.$$
Part d
Similarly to part (a), $$\Delta f \approx \partial_y f \cdot \Delta y.$$
Part e
Because a change in $a$ causes both a change in $x$ and a change in $y$, the resulting change in $f$ will be $$\Delta f \approx \partial_x f \cdot \Delta x + \partial_y f \cdot \Delta y.$$
Plugging in $\Delta x \approx \partial_a x \cdot \Delta a$ and $\Delta y \approx \partial_a y \cdot \Delta a$, we get $$ \Delta f \approx (\partial_x f \cdot \partial_a x + \partial_y f \cdot \partial_a y) \Delta a $$
