# nth Derivative: Definition, Formula, Properties, Examples

The nth derivative of a function is obtained by the successive differentiation of the same function till n times. 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.

## 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).

## nth Derivative Formula

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

1. In the first step, we need to find some derivatives (first, second, third order derivatives, and so on) using the rules of differentiation.
2. From these derivatives, try to find a pattern.
3. Write a function (involving n probably) that follows the above pattern.
4. 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.

Basic concepts of Derivative

Derivative: Definition, Formula, Properties

## nth Derivative Examples

Example 1: Find the nth derivative of ex.

Let f(x)=ex.

Then $f'(x)=e^x$, $f^”(x)=e^x$.

Observe that every time we will get ex when we differentiate ex. Thus, we can conclude that the nth derivative of ex is ex. In other words,

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

Example 2: Find the nth derivative of xn.

Let f(x)=xn.

Then by the power rule of derivatives, we have that

$f’(x)$ = nxn-1

$f’’(x)$ = n(n-1)xn-2

$f’’’(x)$ = n(n-1)(n-2)xn-3

$\vdots$

$f^{(n)}(x)$ = n(n-1)(n-2) . 3 . 2 . 1 . x0 = n! as we know that x0=1.

Thus, the nth derivative of xn 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.

## 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 xn?

Answer: The n-th derivative of xn is equal to n!.

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