The derivative of xn is equal to nxn-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}.
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,
$\dfrac{d}{dx}(x^3)$ = 3x3-1 = 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 nxn-1.
Also Read:
Derivative of root x + 1 by root x
FAQs
Q1: What is the derivative of xn?
Answer: The derivative of xn is equal to nxn-1.