# Definitions > [!NOTE] Definition (Differentiable function over Euclidean spaces) > Let $U \subset \mathbb{R}^n$ be [[Open Sets|open]] and $a\in U$. We say that a [[Function|function]] $f:U \to \mathbb{R}^m$ is **differentiable** at $u$ if there exists a [[Linear Map|linear map]] $L_{a}:\mathbb{R}^n \to \mathbb{R}^m$ such that for all $h\in \mathbb{R}^n$ with $a+h\in U$ we have $f(a+h)=f(a)+L_{a}(h)+R(a,h)$where $\lim_{\substack{ h \to 0 \\ h \neq 0}} \frac{\lVert R(a,h) \rVert}{\lVert h \rVert } =0. $We use $Df(a)$ to denote $L_{a}\in L(\mathbb{R}^n, \mathbb{R}^m)$, which we call the **derivative** of $f$ at $a$. > [!Definition] Definition (Differentiable real function) > A [[Real Function|univariate real-valued function]] $f:I\subseteq \mathbb{R} \to \mathbb{R}$ at $x=c\in I$ is differentiable if its [[Derivative of Real Function|derivative]] given by the [[Limit of Real Function at a Point|limit]]: $\frac{d}{dt}f(c)=f'(c)=\lim_{ \Delta x \to 0 } \frac{f(c+\Delta x)-f(c)}{\Delta x} = \lim_{ x \to c } \frac{f(x)-f(c)}{x-c} $exists. > > > [!NOTE] Definition (Differentiable real-valued function of several real variables) > Let $f:I\to \mathbb{R}$ be a [[Vector-Valued Function of Real variable]]. Let $c\in I.$ Then $f$ is differentiable at $c$ if its component functions are [[Differentiablity|differentiable]]: that is, the [[Limit of Real Function at a Point|limits]] $\begin{align} > \underline{r}'(c) &= \lim_{ h \to 0 } \frac{\underline{r}(c+h)-\underline{r}(c)}{h} \\ > &= \left( \lim_{ h \to 0 } \frac{r_{1}(c+h)-r_{1}(c)}{h}, \dots, \lim_{ h \to 0 } \frac{r_{n}(c+h)-r_{n}(c)}{h} \right) \\ > & = (r_{1}'(c),\dots,r_{n}'(c)) > \end{align}$exist. # Properties **Differentiable real functions**: Note that [[Differentiablity implies Continuity|differentiability implies continuity]]. However continuity doesn't not imply differentiability (e.g [[Weierstrass function]]). We can use [[Mean Value Theorem (Existence of Point at Which Tangent of Arc is Parallel to Secant Through Endpoints)]] to show that [[Positive Derivative Implies Strictly Increasing Real Function]] & [[Real Function with Zero Derivative is Constant]]. See [[Extrema and Derivatives]]. Note that the [[Derivative of Inverse of Strictly Monotonic Differentiable Real Function|inverse of differentiable univariate real-valued function is also differentiable]]. See [[Derivative of Monomials]]. By [[Linearity of Derivative of Differentiable Vector-Valued Function of Single Real Variable]], # Applications - [[Partial Derivatives (of Real-Valued Function on Real n-Space)]]; - [[Antiderivative]]. - [[Differential Equation]]; - [[Power Series is Termwise Differentiable within Radius of Convergence]].