Question

How do you find the nth derivative of the function $f\left( x \right) = {x^n}$?

Answer 1

Hint: In order to determine the nth order derivative, first find out some derivative of the given function up to the order of 3. You will see some patterns are following in the result. Now generalise the pattern by combining the term and in the end use the knowledge of factorial to obtain the simplified result.

Formula used:
$\dfrac{d}{{dx}}({x^n}) = (n){x^{n - 1}}$

Complete step by step answer:
To determine the nth derivative, let’s find out some derivatives
Let $f\left( x \right)$ be y
1st derivative with respect to x
$ \Rightarrow f'\left( x \right) = n{x^{n - 1}}$
2nd derivative with respect to x
$ \Rightarrow f''\left( x \right) = n(n - 1){x^{n - 1}}$
3rd derivative with respect to x
$ \Rightarrow f'''\left( x \right) = n(n - 1)(n - 2){x^{n - 1}}$
So, there is a pattern follow in every derivative we are calculating
The pattern will go on until $n - k = 0$,here k is denoting the order of the derivative. So we can generalise this pattern as
$ \Rightarrow {f^{(k)}}\left( x \right) = n(n - 1)(n - 2)........(n - k + 1){x^{n - k}}$
When we put as $n = k$, then our expression becomes
$
\Rightarrow {f^{(n)}}\left( x \right) = n(n - 1)(n - 2)........(n - n + 1){x^{n - n}} \\
\Rightarrow {f^{(n)}}\left( x \right) = n(n - 1)(n - 2)........(1){x^0} \\
 $
As we know any number raised to the power zero is equal to 1. So ${x^0} = 1$
\[
\Rightarrow {f^{(n)}}\left( x \right) = n(n - 1)(n - 2)........(1){{{x}}^0} \\
\Rightarrow {f^{(n)}}\left( x \right) = n(n - 1)(n - 2)........(1) \\
 \]
\[ n(n - 1)(n - 2)........(1)\] is nothing but equal to $n!$
\[ \Rightarrow {f^{(n)}}\left( x \right) = n!\]
Therefore, the nth order derivative of $f\left( x \right) = {x^n}$ is equal to \[{f^{(n)}}\left( x \right) = n!\].

Additional Information:
1. Calculus consists of two important concepts one is differentiation and other is integration.
2. What is Differentiation?
It is a method by which we can find the derivative of the function. It is a process through which we can find the instantaneous rate of change in function based on one of its variables.
Let y = f(x) be a function of x. So, the rate of change of $y$ per unit change in $x$ is given by:
$\dfrac{{dy}}{{dx}}$.
3. Logarithmic differentiation: It is a way to find the derivatives of complex functions which uses logarithms to find the derivatives. Some cases, differentiating the original function is more difficult than finding the derivative using logarithm. In these cases logarithmic differentiation is used.

Note: Factorials of proper fractions or negative integers are not defined. Factorial n defined only for whole numbers. Derivative is the inverse of integration. Any number raised to the power zero is equal to one.