The **derivative of the function** $f(x)$ with respect to the variable $x$ is the function $f^{\prime}$ whose value at $x$ is $ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} $ provided the limit exists. The derivate can also be expressed as $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ To find $f^{\prime}$ from $f$ is to **differentiate** $f$. ### Notations $y=f(x) \implies f^{\prime}(x)=y^{\prime}=\frac{d y}{d x}=\frac{d f}{d x}=\frac{d}{d x} f(x)=D(f)(x)=D_{x} f(x)$ #### Leibniz $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$ ### Differentiable If $f^{\prime}$ exists at a particular $x$, we say that $f$ is **differentiable** (has a derivative) at $x.$ If $f^{\prime}$ exists at every point in the domain of $f$, we call $f$ **differentiable**. A function $y=f(x)$ is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval. It is differentiable on a closed interval $[a, b]$ if it is differentiable on the interior $(a, b)$ and if the limits $\begin{array}{ll}\lim _{h \rightarrow 0^{+}} \frac{f(a+h)-f(a)}{h} & \text { Right-hand derivative at } a \\ \lim _{h \rightarrow 0^{-}} \frac{f(b+h)-f(b)}{h} & \text { Left-hand derivative at } b\end{array}$ #### Visual cues for Differentiability A function $f$ is not differentiable at a point $P$ (with nearby point Q) if the graph of $f$ at $P$ has 1. a corner, where the one-sided derivatives differ. 2. a cusp, where the slope of $P Q$ approaches $\infty$ from one side and $-\infty$ from the other. 3. a vertical tangent, where the slope of $P Q$ approaches $\infty$ from both sides or approaches $-\infty$ from both sides. 4. a [[Removable Discontinuity]] or a [[‡ Jump Discontinuity]] ### More A collection of derivatives of functions is provided in [[Table of Derivatives]]