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.

Also Read:

Basic concepts of Derivative

Derivative: Definition, Formula, Properties

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 xⁿ.

Answer:

Let f(x)=xⁿ.

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

$\vdots$

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

Also Read:

Derivative of esin x	The derivative of esinx is cosx esinx
Derivative of 1/x	The derivative of 1/x is -1/x2.
Derivative of 1/x2	The derivative of 1/x2 is -2/x3.
Derivative of 1/x3	The derivative of 1/x3 is -3/x4.

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.

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