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: $y = f \left({x}\right)$.

Then the derivative of $y$ with respect to $x$ is defined as:
 * $\displaystyle y^\prime = \lim_{h \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h = D_x f \left({x}\right)$

This is frequently abbreviated as derivative of $y$ WRT $x$.

We introduce the quantity $\delta y = f \left({x + \delta x}\right) - f \left({x}\right)$.

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:
 * $\displaystyle \frac{\mathrm d y}{\mathrm d x} := \lim_{\delta x \to 0} \frac {f \left({x + \delta x}\right) - f \left({x}\right)} {\delta x} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x}$

Hence the notation $f^\prime \left({x}\right) = \dfrac{\mathrm d y}{\mathrm d x}$. 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.