The nth derivative of a function is obtained by the successive differentiation of the same function till n times. The n-th differentiation is referred to the higher order derivatives. In this article, we will learn the definition of the nth derivative along with its formulas, properties, and examples.

Table of Contents

## nth Derivative Definition

Let f(x) be a differentiable function. Its first order derivative of f(x), denoted by f$’$(x), is given by the following limit:

$f'(x)=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$.

If $f’$(x) is differentiable, then the second order derivative of f(x), denoted by $f^”$(x) is determined by the limit

$f^”(x)=\lim\limits_{h \to 0} \dfrac{f'(x+h)-f'(x)}{h}$.

In this way, one can obtain the third, fourth, fifth order derivatives of f(x), and so on. If the (n-1)th derivative $f^{(n-1)}(x)$ is differentiable, then the nth derivative of f(x), denoted by $f^{(n)}(x)$, is defined by the limit below:

$f^{(n)}(x)$ $=\lim\limits_{h \to 0} \dfrac{f^{(n-1)}(x+h)-f^{(n-1)}(x)}{h}$ **…(*)**

**Definition of nth Derivative:** If the functions $f(x), f'(x), \cdots, f^{(n-1)}(x)$ all are differentiable, then the above limit **(*)** is called the nth derivative of f(x).

**Also Read:** Basic concepts of Derivative

Derivative: Definition, Formula, Properties

## nth Derivative Formula

To find the formula for the nth derivative of a function f(x), one has to follow the below steps:

- In the first step, we need to find some derivatives (first, second, third order derivatives, and so on) using the rules of differentiation.
- From these derivatives, try to find a pattern.
- Write a function (involving n probably) that follows the above pattern.
- The function in the third step will be the nth derivative of f(x).

Using these formulas, let us now find a few nth derivatives.

**Related Topics:**

## nth Derivative Examples

Example 1: Find the nth derivative of e^{x}. |

Answer:

Let f(x)=e^{x}.

Then $f'(x)=e^x$, $f^”(x)=e^x$.

Observe that every time we will get e^{x} when we differentiate e^{x}. Thus, we can conclude that the nth derivative of e^{x} is e^{x}. In other words,

$f^{(n)}(x)=e^x$.

Example 2: Find the nth derivative of x^{n}. |

Answer:

Let f(x)=x^{n}.

Then by the power rule of derivatives, we have that

$f’(x)$ = nx^{n-1}

$f’’(x)$ = n(n-1)x^{n-2}

$f’’’(x)$ = n(n-1)(n-2)x^{n-3}

$\vdots$

$f^{(k)}(x)$ = n(n-1)(n-2) . 3 . 2 . [n-(k-1)] . x^{n-k} is the k-th derivative of f(x)=x^{n}.

So to get the n-th derivative, let us put k=n.

Therefore, $f^{(n)}(x)$ = n(n-1)(n-2) . 3 . 2 . 1 . x^{0} = n! as we know that x^{0}=1.

Thus, the nth derivative of x^{n} is equal to n!.

## nth Derivative Properties

The nth derivative of a function satisfies the following properties:

- If $f^{(n-1)}(x)$ is not differentiable, then the nth derivative $f^{(n)}(x)$ does not exist.
- The nth derivative of a constant is always zero.
- The nth derivative of 1 is zero.
- nth derivative is very useful to find the Taylor expansion/Maclaurin series of a function.
- nth derivative is also known as the successive differentiation.

**More Topics:** Leibnitz Theorem on Successive Differentiation: Solved Problems

Derivative of e^{sin x} | The derivative of e^{sinx} is cosx e^{sinx} |

Derivative of 1/x | The derivative of 1/x is -1/x^{2}. |

Derivative of 1/x^{2} | The derivative of 1/x^{2} is -2/x^{3}. |

Derivative of 1/x^{3} | The derivative of 1/x^{3} is -3/x^{4}. |

## FAQs on nth Derivative

**Q1: What is an n-th derivative?**

Answer: The n-th derivative of a function f(x) is the n-th order derivative of f(x). It is the first-order derivative of the (n-1)-th derivative of f(x).

**Q2: What is the n-th derivative of x**

^{n}?Answer: The n-th derivative of x^{n} is equal to n!.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.