📂Analysis

Second Derivative, Higher Order Derivative

Definition¹

Let the function $f$ have a derivative $f^{\prime}$ on the interval $I$. If $f^{\prime}$ itself also has a derivative, we call it the second derivative of $f$ and denote it by $f^{\prime\prime}$.

If $f^{\prime\prime}$ also has a derivative, we denote it as $f^{\prime \prime \prime}$, or simply as $f^{(3)}$. In the same manner, the $n$th derivative of $f$ is denoted as $f^{(n)}$.

$$f,\ f^{\prime},\ f^{\prime\prime},\ f^{(3)},\ \dots,\ f^{(n)}$$

Explanation

Note that it is denoted as $f^{(n)}$ rather than $f^{n}$. Generally, if $n \ge 3$, it is called a higher order derivative.