Leibnitz

Summary

Leibniz's notation for the derivative, which is used in several slightly different ways.

If $y$ is a variable that depends on $x$, then $\textstyle \frac{\mathrm{d}y}{\mathrm{d}x}$, read as "d $y$ over d $x$" (commonly shortened to "d $y$ d $x$"), is the derivative of $y$ with respect to $x$.

If $f$ is a function of a single variable $x$, then $\textstyle \frac{ \mathrm{d} f }{ \mathrm{d}x }$ is the derivative of $f$, and $\textstyle \frac{ \mathrm{d} f }{ \mathrm{d}x }(a)$ is the value of the derivative at $a$. Sometimes $\textstyle \frac{ \mathrm{d} }{ \mathrm{d}x }$ is used as a symbol for the derivative operator, which can be prepended to a function: $\textstyle \frac{ \mathrm{d} }{ \mathrm{d}x } f(x)$.

Total derivative: If $f(x_1, \ldots, x_n)$ is a function of several variables that depend on $x$, then $\textstyle \frac{\mathrm{d}f}{\mathrm{d}x}$ is the derivative of $f$ considered as a function of $x$. That is, $\textstyle\ \frac{\mathrm{d}f}{\mathrm{d}x} = \sum_{i=1}^n \frac{ \partial f }{ \partial x_i }\ \frac{\mathrm{d}x_i}{\mathrm{d}x} ~.$