Partial Derivatives of Functions from Rn To R

Definition: Let $S \subseteq \mathbb{R}^n$ and let $f : S \to \mathbb{R}$. Let $\mathbf{c} = (c_1, c_2, ..., c_n)$ be equal coordinate-wise to $\mathbf{x} = (x_1, x_2, ..., x_n)$ except that $x_k \neq c_k$. The Partial Derivative of $f$ with respect to the $k^{\mathrm{th}}$ variable ($k \in \{ 1, 2, ..., n \}$) is $\displaystyle{\frac{\partial f}{\partial x_k} (\mathbf{c}) = \lim_{x_k \to c_k} \frac{f(x_1, x_2, ..., x_n) - f(c_1, c_2, ..., c_n)}{x_k - c_k}}$ provided that this limit exists.

Other notations for the partial derivative of $f$ with respect to the $k^{\mathrm{th}}$ variable include $D_k f(\mathbf{c})$ (differential operator notation), and $f_k(\mathbf{c})$.

For example, consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined by:

(1)

\begin{align} \quad f(x, y) = x^2 + y^2 \end{align}

Then the partial derivative of $f$ at $\mathbb{c} = (0, 1)$ with respect to the variable $x$ is:

(2)

\begin{align} \quad \frac{\partial}{\partial x} f(0, 1) &= \lim_{x \to 0} \frac{f(x, 1) - f(0, 1)}{x - 0} \\ &= \lim_{x \to 0} \frac{(x^2 + 1^2) - (0^2 + 1^2)}{x - 0} \\ &= \lim_{x \to 0} \frac{x^2}{x} \\ &= \lim_{x \to 0} x \\ &= 0 \end{align}

The partial derivative of $f$ at $\mathbf{c} = (0, 1)$ with respect to the variable $y$ is:

(3)

\begin{align} \quad \frac{\partial}{\partial y} f(0, 1) &= \lim_{y \to 1} \frac{f(0, y) - f(0, 1)}{y - 1} \\ &= \lim_{y \to 1} \frac{(0^2 + y^2) - (0^2 + 1^2)}{y - 1} \\ &= \lim_{y \to 1} \frac{y^2 - 1}{y - 1} \\ &= \lim_{y \to 1} y + 1 \\ &= 2 \end{align}

So for a function of $n$ variables there are at most $n$ partial derivatives of $f$ at any particular point $\mathbf{c}$ in the domain of $f$.

That said, though the term "derivative" is used to denote partial derivatives of a function, they are actually in a sense weaker a weaker concept than the derivative of a single-variable function $f : \mathbb{R} \to \mathbb{R}$. We will look more into this shortly.