Notations

Here are the four main ones: $f'$, $f'(x)$, $\dfrac{df}{dx}$, and $\dfrac{df(x)}{dx}$. These are purely stylistic differences.


Note: $\dfrac{df}{dx}$ can also be written as $\dfrac{d}{dx}f$. This notation is sometimes used when someone wants to emphasize that $\dfrac{d}{dx}$ is a mathematical operator.

Again, here are the four main ones: $f''$, $f''(x)$, $\dfrac{d^2 f}{dx^2}$, and $\dfrac{d^2 f(x)}{dx^2}$. These are purely stylistic differences.


Note: $\dfrac{d^2 f}{dx^2}$ can also be written as $\left(\dfrac{d}{dx}\right)^2\!f~$ or $~\dfrac{d^2}{dx^2}f$. This style is sometimes used when someone wants to emphasize that $\dfrac{d}{dx}$ is a mathematical operator.

Here are the main ones: $f'(a)$, $f'(x)|_{x=a}$, $f'(x)|_a$, $\left.\dfrac{df}{dx}\right|_{x=a}$, and $\left.\dfrac{df(x)}{dx}\right|_{x=a}$.

Here are the main ones: $f''(a)$, $f''(x)|_{x=a}$, $f''(x)|_a$, $\left.\dfrac{d^2 f}{dx^2}\right|_{x=a}$, and $\left.\dfrac{d^2 f(x)}{dx^2}\right|_{x=a}$.