#todo > [!NOTE] Definiton > Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a [[Real-Valued Function on Real n-Space (Multivariable Function)|multivariable function]]. The 'direction of steepest descent' of $f(\underline{x})$ is $-\nabla f$ where $\nabla f$ denotes the [[Fréchet Differentiation|gradient]] of $f.$ **Informal proof**: By definition of [[Directional Derivatives|directional derivative]], $D_{\underline{u}}=\nabla f \cdot \underline{u} = |\nabla f|\cos \theta$by [[Dot Product in Real n-Space in Terms of Angle & Length]], where $\theta\in[0,\pi]$ is the acute angle $\angle \nabla f \;\underline{u}.$ Thus the directional derivative is maximised when $\theta =0,$ that is $\underline{u}$ and $\nabla f$ are parallel, so $f$ decreases the fastest when we walk in the direction $\underline{u}=-\nabla f/|\nabla f|.$