Definition:Differential/Real Function/Point

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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 $\rd f \left({x}\right) : \R \to \R$ defined as:

$\rd f \left({x}\right) \left({h}\right) = f' \left({x}\right) \cdot h$

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