# Definitions > [!NOTE] Definition (Directional Derivative) > Let $f: U \subset \mathbb{R}^{n}\to \mathbb{R}$ be a [[Real-Valued Function on Real n-Space (Multivariable Function)|multivariable function]]. Let $\underline{u}\in \mathbb{R}^{n}$ be a [[Real n-Space|real vector]] of [[Euclidean Norm|length]] $1.$ The directional derivative of $f$ in the direction of $\underline{u}$ is defined by the [[Limit of Real Function at a Point|limit]]: $D_{u}f(\underline{x}) = \lim_{ h \to 0 } \frac{f(\underline{x}+h\underline{u})-f(\underline{x})}{h}$ # Properties # Example Suppose the temperature at point $(x, y)$ on the football pitch is given by $f (x, y) = 10x - x^{2}y$. What is the rate of change of temperature if you walk in the direction $\underline{u}=\underline{i}+\underline{j}$? *Solution*. Compute $\nabla f = (10-2xy , -x^{2})^{T}$ so $\nabla f(1,1)=(8,-1)^{T}$. Also $\hat{\underline{u}}= \frac{1}{\sqrt{ 2 }} (1,1)^{T}$. So $D_{\underline{u}}f(1,1)= \frac{1}{\sqrt{ 2 }} (1,1)^{T} \cdot(8,-1)^{T}= \frac{7}{\sqrt{ 2 }}$.