The derivative of x^{n} is equal to nx^{n-1}. The function x^{n} is read as x to the power n. The derivative of x to the n is referred to as the power rule of derivatives. In this post, we will find the derivative of x^{n} by the limit definition of derivatives and the power rule. In the end, we will provide a few applications.

## Derivative of x^{n} by First Principle

Let us first recall the first principle of derivatives. It says that the derivative of a function f(x) by the first principle is given by the following limit formula:

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

Let $f(x)=x^n$ in the above limit. Thus, $f(x+h)=(x+h)^n$.

So the derivative of x to the n by the first principle will be as follows.

$\dfrac{d}{dx}(x^n)$ $=\lim\limits_{h \to 0} \dfrac{(x+h)^n-x^n}{h}$

Let us put $z=x+h$. Thus $z \to x$ as $h \to 0$. Also, $h=z-x$.

$\therefore \dfrac{d}{dx}(x^n)$ $=\lim\limits_{z \to x} \dfrac{z^n-x^n}{z-x}$

$=nx^{n-1}$ using the limit formula $\lim\limits_{x \to a} \dfrac{x^n-a^n}{x-a}=na^{n-1}$

Hence, the derivative of x^n by the first principle is nx^{n-1}.

**Remark:** The second derivative of x^{n} is obtained by differentiating its first derivative nx^{n-1} which is equal to n(n-1)x^{n-2}.

Question 1: Find the derivative of x cube , that is, find $\dfrac{d}{dx}(x^3)$ |

*Solution:*

In the above formula, that is, $\dfrac{d}{dx}(x^n)=nx^{n-1}$, we will put $n=3$ in order \to get the derivative of x cube. Thus,

$\dfrac{d}{dx}(x^3)$ = 3x^{3-1} = 3x^{2}.

Thus, the derivative of x^{3} is equal to 3x^{2}.

Next, we will find out the derivative of x to the power n by the power rule of derivatives.

## Derivative of x^{n} by Power Rule

To find the derivative of x to the n using the power rule of derivatives, let us recall the power rule of derivatives, and the rule is given below.

$\dfrac{d}{dx}(x^k)=kx^{k-1}$

Putting k=n in the above rule, we will get the derivative of x^{n}. Hence, it follows that

$\dfrac{d}{dx}(x^n)=nx^{n-1}$.

Therefore, the derivative of x^{n} by the power rule of derivatives is nx^{n-1}.

**Also Read:**

**Derivative of root x + 1 by root x**

## FAQs

**Q1: What is the derivative of x ^{n}?**

**Answer:** The derivative of x^{n} is equal to nx^{n-1}.

**Q1: What is the second derivative of x ^{n}?**

**Answer:** The second derivative of x^{n} is equal to n(n-1)x^{n-2}.