\begin{definition}[Definition:Derivative/Real Function/With Respect To]
Let $f$ be a real function which is differentiable on an open interval $I$.
Let $f$ be defined as an equation: $y = \map f x$.
Then the '''derivative of $y$ with respect to $x$''' is defined as:
:$\ds y^\prime = \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h = D_x \, \map f x$
This is frequently abbreviated as '''derivative of $y$''' '''WRT''' or '''w.r.t.''' '''$x$''', and often pronounced something like '''wurt'''.
We introduce the quantity $\delta y = \map f {x + \delta x} - \map f x$.
This is often referred to as '''the small change in $y$ consequent on the small change in $x$'''.
Hence the motivation behind the popular and commonly-seen notation:
:$\ds \dfrac {\d y} {\d x} := \lim_{\delta x \mathop \to 0} \dfrac {\map f {x + \delta x} - \map f x} {\delta x} = \lim_{\delta x \mathop \to 0} \dfrac {\delta y} {\delta x}$
Hence the notation $\map {f^\prime} x = \dfrac {\d y} {\d x}$.
This notation is useful and powerful, and emphasizes the concept of a derivative as being the limit of a ratio of very small changes.
However, it has the disadvantage that the variable $x$ is used ambiguously: both as the point at which the derivative is calculated and as the variable with respect to which the derivation is done.
For practical applications, however, this is not usually a problem.
\end{definition}