Differentiation and Derivative

\[ \frac{\partial f}{\partial x_{j}} (\mathbf{a}) = \lim_{h\to 0}\frac{f(\mathbf{a}+h\mathbf{e}_{j}) - f(\mathbf{a})}{h} \]

Let \(f:X\mapsto\mathbb{R}\) be a function where \( X\subset \mathbb{R}^{n} \) is open. Suppose \( f_{x_{j}}(\mathbf{a}) \) exists for \( j=1,2,\dots,n \) where \( \mathbf{a}\in X \). Define

\[ h(\mathbf{x}) = f(\mathbf{a}) + \begin{bmatrix}f_{x_{1}}(\mathbf{a}) \\ f_{x_{2}}(\mathbf{a}) \\ \vdots \\ f_{x_{n}}(\mathbf{a})\end{bmatrix} \cdot (\mathbf{x} - \mathbf{a}) \]

We say \( f \) is differentiable at \(\mathbf{a}\) if

\[ \lim_{\mathbf{x}\to\mathbf{a}} \frac{f(\mathbf{x}) - f(\mathbf{a})}{\mathbf{x}-\mathbf{a}} = 0 \]

We say \( f \) is differentiable if it is differentiable at every point in \(X\).

Intuitively, we can interpret differentiability as the function can be “approximated” by some linear function.

Theorem. Let \( f:X\mapsto\mathbb{R} \) be a function where \( X\subset\mathbb{R}^{n} \), and let \( \mathbf{a} \) be an interior point of \(X\). If \(f\) has continuous partial derivatives in an open set containing \(\mathbf{a}\), then \(f\) is differentiable at \(\mathbf{a}\).

Proposition. If \(f\) is differentiable at \(\mathbf{a}\), then \(f\) is continous at \(\mathbf{a}\).

Let \( f:X\mapsto\mathbb{R} \) be a function where \( X\subset\mathbb{R}^{n} \), and let \( \mathbf{a} \) be an interior point of \( X \). Let \( \mathbf{u} \) be a unit vector in \( \mathbb{R}^{n} \). The directional derivative of \( f \) at \( \mathbf{a} \) in the direction of \( \mathbf{u} \) is defined by

\[ D_{\mathbf{u}} f(\mathbf{a}) = \lim_{h\to 0}\frac{f(\mathbf{a}+h\mathbf{u}) - f(\mathbf{a})}{h} \]

Proposition. Suppose \( f \) is differentiable at \( \mathbf{a} \). Then for any unit vector \( \mathbf{u} \), the directional derivative at \( \mathbf{a} \) in the direction of \( \mathbf{u} \) exists and

\[ D_{\mathbf{u}} f(\mathbf{a}) = (\nabla f)(\mathbf{a}) \cdot \mathbf{u} \]