Differentiability and the Differential of a Function at a Point: A function \(f: E \to \mathbb{R}^n\) defined on a set \( E \subset \mathbb{R}^m \) is differentiable at the point \( x \in E \), which is a limit point of \( E \), if \[\boxed{f(x + h) - f(x)= L(x) h + \alpha(x; h)}\] where \( L(x): \mathbb{R}^m \to \mathbb{R}^n \) is a linear function in \( h \), and \(\alpha(x; h) = o(h) \text{ as } h \to 0, x + h \in E\)

The Differential and Partial Derivatives of a Real-Valued Function: If the vectors \( f(x + h), f(x), L(x)h, \alpha(x; h) \) in \( \mathbb{R}^n \) are written in coordinates, \(\boxed{f(x + h) - f(x)= L(x) h + \alpha(x; h)}\) becomes equivalent to the \( n \) equalities \[ f^i(x + h) - f^i(x) = L^i(x) h + \alpha^i(x; h), \quad (i = 1, \dots, n) \]

If a function \(f: E \to \mathbb{R}\) defined on a set \( E \subset \mathbb{R}^m \) is differentiable at an interior point \( x \in E \), then the function has a partial derivative at that point with respect to each variable, and the differential of the function is uniquely determined by these partial derivatives in the form \[ \mathrm{d}f(x)h = \frac{\partial f}{\partial x^1}(x) h^1 + \cdots + \frac{\partial f}{\partial x^m}(x) h^m \]