Definition:Differential of Mapping/Real Function/Point

Definition
Let $U\subset\R$ be an open set.

Let $f : U \to \R$ be a real function.

Let $f$ be differentiable at a point $x \in U$.

The differential of $f$ at $x$ is the linear transformation $\mathrm d f(x) : \R \to \R$ defined as:
 * $\mathrm d f(x) (h) = f'(x) \cdot h$

where $f' \left({x}\right)$ is the derivative of $f$ at $x$.