Note that n-th root can be written as raised to the power 1/n}. The derivative of n-th root of x is 1/n x^{1/n -1}. In this post, we will find the derivative of n-th root of x.
Remark: The nth root of x is written as
Derivative of nth root of x
Using the rule of indices, we can write nth root of x as $x^{\frac{1}{n}}$. As it is a power of $x$, its derivative can be computed by the power rule of derivatives.
Power Rule of Derivative: Recall, the power rule of derivatives.
$\dfrac{d}{dx}(x^m)=mx^{m-1}$ $\quad \cdots$ (i)
Putting $m=\dfrac{1}{n}$ in the above formula (i), we will get the derivative of nth root of x. Thus
$\dfrac{d}{dx}(x^{\frac{1}{n}})$
$=\dfrac{1}{n} x^{\frac{1}{n}-1}$
$=\dfrac{1}{n} x^{\frac{1-n}{n}}$
$=\dfrac{1}{n} x^{\frac{1}{n}(1-n)}$
Simplifying the above expression by the law of indices, we obtain that
$\dfrac{d}{dx}(x^{\frac{1}{n}})$ $=\dfrac{1}{n} x^{-\frac{1}{n}(n-1)}$ $=\dfrac{1}{n} x^{\frac{1}{n}(n-1)}$ $=\dfrac{1}{n} \sqrt[n]{x^{n-1}}$
Thus, the derivative of n-th root of x is equal to $\dfrac{1}{n} \sqrt[n]{x^{n-1}}$.
Remark:
(1) If we put n=2, then nth root means square root. Putting n=2 in the above, we obtain the derivative of root x, which is
$\dfrac{d}{dx}(\sqrt{x})$ $=x^{\frac{1}{2}(2-1)}/2$ $=x^{\frac{1}{2}}/2$ $=\sqrt{x}/2$
(2) Similarly, if we put n=3, 4, then we will get the derivatives of the cube (third) and fourth root of x respectively.
Also Read:
Derivative of Fourth Root of x
FAQs
Q1: What is the derivative of nth root x?
Answer: The derivative of nth root x is $\dfrac{1}{n} \sqrt[n]{x^{n-1}}$.