Partial Differentiability (Real Vector Functions)#

Definition: Partial Differentiability

Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function and let \(\boldsymbol{p}\) and let be an interior point of \(\mathcal{D}\).

We say that \(f\) is partially differentiable at \(\boldsymbol{p}\) w.r.t. the \(k\)-th variable (\(k \in \{1, \dotsc, n\}\)) if \(f\) is directionally differentiable at \(\boldsymbol{p}\) along the \(k\)-th standard basis vector \(\boldsymbol{e}_k\).

Definition: Partial Derivative

The corresponding directional derivative is known as \(f\)'s partial derivative at \(\boldsymbol{p}\) w.r.t. \(k\)-th variable:

\[\lim_{h \to 0} \frac{f(\boldsymbol{p} + h\boldsymbol{e}_k) - f(\boldsymbol{p})}{h}\]

Notation

In general, the partial derivative of \(f\) at \(\boldsymbol{p}\) w.r.t. the \(k\)-th variable is denoted as follows:

\[\partial_k f(\boldsymbol{p})\]

If labels (for example \(x_1, \dotsc, x_n\)) are introduced for the components of \(\boldsymbol{p}\), we also use the following notations:

\[\partial_{x_k}f(x_1, \dotsc, x_n) \qquad \partial_{x_k}f(x_1, \dotsc, x_n) \qquad \frac{\partial f}{\partial x_k}(x_1, \dotsc, x_n)\]

The labels \(x, y\) and \(x, y, z\) are very common for \(\mathbb{R}^2\) and \(\mathbb{R}^3\), respectively.

We also use the term partial derivative for each real vector function which maps each \(\boldsymbol{p}\) to \(f\)'s respective partial derivative.

Theorem: Partial Differentiability via Component Functions

Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function and let \(\boldsymbol{p}\) be an interior point of \(\mathcal{D}\).

Then \(f\) is partially differentiable at \(\boldsymbol{p}\) w.r.t. the \(k\)-th variable if and only if all of its component functions \(f_1, \dotsc, f_n\) are partially differentiable at \(\boldsymbol{p}\) w.r.t. the \(k\)-th variable. In this case:

\[\partial_{k} f(\boldsymbol{p}) = \begin{bmatrix} \partial_{k}f_1 (\boldsymbol{p}) \\ \vdots \\ \partial_{k}f_n (\boldsymbol{p})\end{bmatrix}\]

Proof

TODO

Theorem: Chain Rule for Partial Derivatives

Let \(g: \mathcal{D}_g \subseteq \mathbb{R}^l \to \mathbb{R}^m\) and \(f: \mathcal{D}_f \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be real vector functions with component functions \(g_1, \dotsc, g_m\) and \(f_1, \dotsc, f_n\), respectively. Let \(\boldsymbol{p}\) be an interior point of \(\mathcal{D}_g\) such that \(g(\boldsymbol{p})\) is an interior point of \(\mathcal{D}_f\).

If \(g\) is partially differentiable at \(\boldsymbol{p}\) w.r.t. the \(k\)-th variable (\(k \in \{1, \dotsc, l\}\)) and \(f\) is totally differentiable at \(g(\boldsymbol{p})\), then \(f \circ g\) is partially differentiable at \(\boldsymbol{p}\) w.r.t. the \(k\)-th variable:

\[\partial_k (f \circ g)(\boldsymbol{p}) = \begin{bmatrix} \sum_{j=1}^m \partial_j f_1(g(\boldsymbol{p})) \partial_k g_j(\boldsymbol{p}) \\ \vdots \\ \sum_{j=1}^m \partial_j f_n(g(\boldsymbol{p})) \partial_k g_j(\boldsymbol{p}) \end{bmatrix}\]

Proof

TODO