The derivative of fourth root of x is 1/(4 x^{3/4}). In this post, we will find the derivative of the fourth root of x using the power rule of derivatives.
Note that fourth root of x can be written as
![Fourth Root of x Fourth Root of x](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjd2O82f6OTyhsLyzkNuyAfqQVj0MJBAEJZocawuG0C4ZVJhCoLOsDcBcwLojfeSoGWEFCYjIxhhrVV3wvabUdUCWbOtfhzt7W2NJaAW9BQ8CzsjrKXF5gEFnN3pLQdydOAWb1DZMlfRI1qwgI5mMX2XW_Drf0kWXpT5jsQ3Eh_8A2T6ZIP0COu1aFE/s16000/fourth%20root%20of%20x.png)
Alternatively, one can write it as $x^{\dfrac{1}{4}}$.
Derivative of Fourth Root of x
To find the derivative of the fourth root of $x$, we will use the power rule of derivatives. The rule says that the derivative of x to the power n (n is an integer) is
$\dfrac{d}{dx}(x^n)=nx^{n-1}$ $\quad \cdots (I)$
Now, by the rule of indices, the fourth root of $x$ is expressed as $x^{\frac{1}{4}}$. So the derivative of the fourth root of $x$ is
$\dfrac{d}{dx}(x^{\frac{1}{4}})$
$=\dfrac{1}{4} x^{\frac{1}{4}-1}$ by the above rule (I) with $n=\dfrac{1}{4}$.
$=\dfrac{1}{4} x^{\frac{-3}{4}}$
$=\dfrac{1}{4x^{3/4}}$
So the derivative of fourth root of x is 1/(4x^{3/4}).
Question 1: Find the derivative of fourth root of x at x=1.
Solution:
We have seen above that the derivative of fourth root of x is 1/(4x^{3/4}). So the derivative of x to the power 1/4 at x=1 will be
$\dfrac{d}{dx}(x^{\frac{1}{4}})|{x=1}$ $=[\dfrac{1}{4x^{3/4}}]{x=1}$ $=\dfrac{1}{4\cdot 1^{3/4}}$ $=\dfrac{1}{4}.$
Question 2: What is a derivative of 4?
Solution:
Note that 4 is a constant function of x. So the derivative of 4 with respect to x will be zero as we know that the Derivative of a constant is 0.
Derivative of Fourth Root of 1
Note that fourth roots of 1 are the solutions of the equation $x^4=1$. These solutions are constants. As we know that the Derivative of a constant is zero, so we can say that the derivative of the fourth root of 1 is zero.
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FAQs
Q1: What is the derivative of fourth root x?
Answer: The derivative of fourth root x is equal to 1/(4x^{3/4}).