>[!quote] Format Definition >"The *instantaneous rate of change* of a function with respect to one of its [[Variable|variables]]" The derivative is a [[Linear Transformation|Linear]] [[Higher-Order]] [[Function]] describing the rate of change of a function along one of its parameters. For $f(x)=y$, the derivative of $f$ ($f'(x)$ or $\frac{df}{dx}$) is equal to: $\huge \frac{df}{dx}(x)= \lim_{h \to \infty} \frac{ f(x+h) - f(x) }{h} $ The notation for the derivative of some function $f(x)$ can be notated as: $\huge \begin{align} \deriv{f}{x} &= \dot f = f' \\ \deriv{^2f}{x^2} &= \ddot f = f'' \\ \deriv{^3f}{x^3} &= \dddot f = f''' \\ \end{align}$ And for the $nth ordered derivative: $\huge \begin{align} \deriv{^n f}{x^n}(x) \\ \deriv{^n \pa{f(x)}}{x^n} \\ \deriv{^n }{x^n}\pa{f(x)} \\ \end{align}$ It is helpful though (and my preferred notation) is to treat the derivative in notation as an [[Operator]] on a function, $\huge \begin{align} \mathcal D^n\{f\} &= \deriv{^n f}{x^n} \end{align}$ This approach is more appealing to me because it leverages consistent notation with [[Functional Powers]], and the curly braces are reminiscent of other forms of calculus such as the [[Laplace Transform]]. You can also be more explicit on what variable you are differentiating, eg: $\huge \mathcal D^n_{x}\{f\} = \deriv{^n f}{x^n} $ If $f$ is not a single variable function however (see [[Partial Derivative]]), it is common to instead use the greek symbol 'del' $\partial$. $\huge \partial^n_{x}\{f\} = \frac{\partial ^n f}{\partial x^n} $