Partial Derivatives and Total Differentials

Let $z = f(x, y)$ be defined on a neighborhood of $(x_0, y_0)$. Fix $y = y_0$ and give $x$ an increment $\Delta x$ at $x_0$. If the limit

$\lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x, y_0) - f(x_0, y_0)}{\Delta x}$

exists, it is called the partial derivative of $f$ with respect to $x$ at $(x_0, y_0)$, written

$f_x(x_0, y_0), \quad \frac{\partial z}{\partial x}\bigg|_{(x_0, y_0)}, \quad \frac{\partial f}{\partial x}\bigg|_{(x_0, y_0)}, \quad z_x\big|_{(x_0, y_0)}$

Similarly, the partial derivative with respect to $y$ is

$f_y(x_0, y_0) = \lim_{\Delta y \to 0} \frac{f(x_0, y_0 + \Delta y) - f(x_0, y_0)}{\Delta y}$

The second partial derivatives of $z = f(x,y)$ are:

$\frac{\partial^2 z}{\partial x^2}, \quad \frac{\partial^2 z}{\partial y^2}, \quad \frac{\partial^2 z}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right), \quad \frac{\partial^2 z}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)$

The mixed ones $\frac{\partial^2 z}{\partial x\partial y}$ and $\frac{\partial^2 z}{\partial y\partial x}$ are called mixed partial derivatives.

Let $z = f(x, y)$ be defined on a neighborhood of $(x_0, y_0)$. If the total increment

$\Delta z = f(x_0 + \Delta x, y_0 + \Delta y) - f(x_0, y_0)$

can be written as

$\Delta z = A\Delta x + B\Delta y + o(\rho)$

where $A, B$ do not depend on $\Delta x, \Delta y$ and $\rho = \sqrt{(\Delta x)^2 + (\Delta y)^2}$, then $f$ is differentiable at $(x_0, y_0)$, and $A\Delta x + B\Delta y$ is its total differential:

$dz = A\Delta x + B\Delta y \quad \text{or} \quad df = A\Delta x + B\Delta y$