The **derivative of a function $f$ at a point $x_{0}$**, denoted $f^{\prime}\left(x_{0}\right)$, is $ f^{\prime}\left(x_{0}\right)=\lim _{h \rightarrow 0} \frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h} $ provided this [[Limit (ε, δ)|limit]] exists. This is also called the **rate of change** of $f(x)$ with respect to $x$ at $x=x_{0}$ ### Notations To indicate the value of a derivative at a specified number $x=a$, we use the notation $ f^{\prime}(a)=\left.\frac{d y}{d x}\right|_{x=a}=\left.\frac{d f}{d x}\right|_{x=a}=\left.\frac{d}{d x} f(x)\right|_{x=a} $ [[Slope of the curve]] [[Tangent line]] [[Difference Quotient]] [[Derivative of a function]]