Derivative of x^n by First Principle, Power Rule

The derivative of xⁿ is equal to nx^n-1. The function xⁿ 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 xⁿ by the limit definition of derivatives and the power rule. In the end, we will provide a few applications.

Derivative of xⁿ 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 xⁿ 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 = 3x².

Thus, the derivative of x³ is equal to 3x².

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

Derivative of xⁿ 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 xⁿ.  Hence, it follows that

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

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

Also Read: 

Derivative of 1/root(x)

Derivative of (sinx)^logx

Derivative of x^2

Derivative of root x + 1 by root x

Derivative of x^3/2

Derivative of 1/(1+x^2)

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