# Derivative of Fourth Root of x | 4√x Derivative

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

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}).