Problem on deriving the chain rule for functions of several variables

Consider the composition of functions $f(x(a,b),y(a,b))$.
1. Keeping $b$ fixed, and varying $a$ by $\Delta a$, what is the approximate change in $x(a,b)$?
2. What is the approximate change in $y(a,b)$?
3. Keeping $y$ fixed, and varying $x$ by $\Delta x$, how much does $f(x,y)$ change?
4. Keeping $x$ fixed, and varying $y$ by $\Delta y$, how much does $f(x,y)$ change?
5. Combine your answers above to compute the partial derivative $\partial_a f(x(a,b), y(a,b))$. This should be the chain rule.
• Solution

Part a

Recall that
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 $$\Delta f \approx (\partial_x f \cdot \partial_a x + \partial_y f \cdot \partial_a y) \Delta a$$