Definition:Derivative/Real Function/With Respect To

Definition

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:

$\displaystyle y^\prime = \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h = D_x \, \map f x$

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:

$\displaystyle \dfrac {\d y} {\d x} := \lim_{\delta x \mathop \to 0} \dfrac {\map f {x + \delta x} - \map f x} {\delta x} = \lim_{\delta x \mathop \to 0} \dfrac {\delta y} {\delta x}$

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.