The derivative of fourth foot of x is equal to 1/(4x^{3/4}). Fourth root of x is denoted by ∜x = x^{1/4}, so its derivative formula is given by

$\dfrac{d}{dx}$(∜x) = $\dfrac{1}{4x^{3/4}}.$

In this article, we will learn how to differentiate fourth root of x with respect to x by the following methods:

- Power rule of derivatives
- Substitution method
- First principle of differentiation
- Logarithmic differentiation.

Table of Contents

## By Power Rule

As the fourth root of x is expressed as ∜x = x^{1/4}, its derivative can be easily computed by the power rule of differentiation. This rule says that the derivative of x^{n} is equal to nx^{n-1}, that is,

d/dx (x^{n}) = nx^{n-1}.

Put n=1/4.

So we obtain that

$\dfrac{d}{dx}$(∜x) = $\dfrac{d}{dx}(x^{1/4})$ ⇒ $\dfrac{d}{dx}$(∜x) = $\dfrac{1}{4}x^{\frac{1}{4}-1}$ ⇒ $\dfrac{d}{dx}$(∜x) = $\dfrac{1}{4x^{3/4}}$. |

So the derivative of fourth root of x by power rule is equal to 1/(4x^{3/4}).

**ALTERNATIVE METHOD:** To find the derivative of fourth root of x, let us substitute $y=\sqrt[4]{x}.$. This implies that

y^{4}=x.

Differentiate both sides with respect to x. So we get that

$4y^3 \dfrac{dy}{dx}=1$

⇒ $\dfrac{dy}{dx}=\dfrac{1}{4y^3}$

⇒ $\dfrac{dy}{dx}=\dfrac{1}{4(\sqrt[4]{x})^3}$ as we have $[\because y=\sqrt[4]{x}]$

⇒ $\dfrac{dy}{dx}=\dfrac{1}{4x^{3/4}}$

This proves that the derivative of fourth root of x is 1/(4x^{3/4}), and we get this by the substitution method.

**Also Read:** Derivative of cube root of x

## Derivative of Fourth Root of x by First Principle

By first principle, the derivative of f(x) is given by the following limit formula:

d/dx (f(x)) = lim_{h→0} $\dfrac{f(x+h)-f(x)}{h}$.

Let f(x) = ∜x. Note that

f(x) = x^{1/4} and f(x+h) = (x+h)^{1/4}.

Then by first principle,

$\dfrac{d}{dx}$(∜x) = lim_{h→0} $\dfrac{(x+h)^{\frac{1}{4}} – x^{\frac{1}{4}}}{h}$ |

Put x+h = z. So z→x as h→0. Also, h=z-x.

Therefore,

So $\dfrac{d}{dx}$(∜x) = lim_{z→x} $\dfrac{z^{\frac{1}{4}} – x^{\frac{1}{4}}}{z-x}$ = $\dfrac{1}{4}x^{\frac{1}{4}-1}$.Here we have used the formula lim _{x→a} (x^{n}-a^{n})/(x-a) = na^{n-1}. |

Hence, $\dfrac{d}{dx}$(∜x) = $\dfrac{1}{4x^{3/4}}$.

Therefore, the derivative of fourth root of x is equal to 1/(4x^{3/4}) and this is obtained by the first principle of differentiation.

**Related Topics:**

## By Logarithmic Differentiation

Let y = ∜x = x^{1/4}.

Taking natural logarithms on both sides, we get that

ln y = 1/4 ln x

Differentiating with respect to x,

$\dfrac{1}{y} \dfrac{dy}{dx}=\dfrac{1}{4} \cdot \dfrac{1}{x}$

⇒ $\dfrac{dy}{dx}=\dfrac{y}{4x} = \dfrac{x^{1/4}}{4x} = = \dfrac{1}{4x^{3/4}}$.

So by logarithmic differentiation, the derivative of fourth root of x is equal to 1/(4x^{3/4}).

## Solved Problems

**Question 1:** Find the derivative of fourth root of a where a is a constant. That is, find d/dx(∜a).

**Answer:**

Note that the fourth root of $a$ is a constant. We know that the derivative of a constant is zero. So we obtain that

d/dx(∜a) = 0.

⇒ d/dx(a^{1/4}) = 0.

**Remark:** Put a=1. So the derivative of fourth root of 1 with respect to x is equal to 0, that is, d/dx(∜1) = 0.

**Question 2:** Find the derivative of fourth root of x+1, i.e, find d/dx(∜(x+1)).

**Answer:**

Put t=x+2.

So dt/dx = 1.

Now by the chain rule of derivatives,

$\dfrac{d}{dx}$ (∜(x+1)) = $\dfrac{d}{dt}$ (∜t) × $\dfrac{dt}{dx}$

= 1/(4t^{3/4}) × 1

= 1/{4(x+1)^{3/4}}.

So the derivative of (x+1)^{1/4}, that is, fourth root of x+1 is equal to 1/{4(x+1)^{3/4}}.

## FAQs

**Q1: What is the derivative of x**

^{1/4}(fourth root of x)?Answer: The derivative of x^{1/4} (fourth root of x) is equal to 1/(4x^{3/4}).

**Q2: What is d/dx(∜x)?**

Answer: d/dx(∜x) = 1/(4x^{3/4}).

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.