Differentiability of Functions of Several Variables

Recall that when we have a function of a single variable, $y = f(x)$, then the increment of $y$ denoted $\Delta y = f(a + \Delta x) - f(a)$ which represents the change in $y$-value as a result of a change of $\Delta x$ in $x$-values.

Let $z = f(x, y)$ be a two variable real-valued function. If $x$ changes from $a$ to $a + \Delta x$, then the difference in $x$-coordinates in is $\Delta x$. If $y$ changes from $b$ to $b + \Delta y$ then the difference in $y$-coordinates is $\Delta y$. Therefore, the difference in $z$-coordinates is given by:

(1)

\begin{align} \Delta z = f(a + \Delta x, b + \Delta y) - f(a, b) \end{align}

So $\Delta z$ is the change when $(x, y)$ changes from $(a, b)$ to $(a + \Delta x, b + \Delta y)$ in the domain of $f$. This value $\Delta z$ has a special name.

Definition: Let $z = f(x, y)$ be a two variable real-valued function. Suppose that $x$ changes from $a$ to $a + \Delta x$ The Increment of $z$ or Change in $z$ is denoted by $\Delta z = f(a + \Delta x, b + \Delta y)$.

We will now define what it means for a two variable function to be differentiable.

Definition: Let $z = f(x, y)$ be a two variable real-valued function. $f$ is said to be Differentiable at $(a, b) \in D(f)$ if $\Delta z = f_x (a, b) \Delta x + f_y (a, b) \Delta y + \epsilon_1 \Delta x + \epsilon_2 \Delta y$ where $\epsilon_1, \epsilon_2 \to 0$ as $\Delta x, \Delta y \to 0$.