Symbols for derivatives

Since $\Delta\, y$ and $\Delta\, x$ are always finite and have definite values, the expression

$$\frac{\Delta y}{\Delta x}$$

is really a fraction. The symbol

$$\frac{dy}{dx},$$

however, is to be regarded not as a fraction but as the limiting value of a fraction. In many cases it will be seen that this symbol does possess fractional properties, and later on we shall show how meanings may be attached to $dy$ and $dx$, but for the present the symbol $\frac{dy}{dx}$ is to be considered as a whole.

Since the derivative of a function of $x$ is in general also a function of $x$, the symbol $f'(x)$ is also used to denote the derivative of $f(x)$.

Hence, if $y = f(x)$, we may write $\frac{dy}{dx} = f'(x)$, which is read ``the derivative of $y$ with respect to $x$ equals $f$ prime of $x$.'' The symbol

$$\frac{d}{dx}$$

when considered by itself is called the differentiating operator, and indicates that any function written after it is to be differentiated with respect to $x$. Thus

         $\frac{dy}{dx}$ or $\frac{d}{dx} y$ indicates the derivative of $y$ with respect to $x$;

         $\frac{d}{dx} f(x)$ indicates the derivative of $f(x)$ with respect to $x$;

         $\frac{d}{dx} (2x^2 + 5)$ indicates the derivative of $2x^2 + 5$ with respect to $x$;

        $y'$ is an abbreviated form of $\frac{dy}{dx}$.

The symbol $D_x$ is used by some writers instead of $\frac{d}{dx}$. If then

$$y = f(x),$$

we may write the identities

$$y' = \frac{dy}{dx} = \frac{d}{dx} y = D_x f(x) = f'(x).$$