# 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** or **w.r.t.** **$x$**, and often pronounced something like **wurt**.

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.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 13$: Definition of a Derivative: $13.1$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 10.4$