Cube Root of x: Definition, Symbol, Graph, Properties, Derivative, Integration

The cube root of x is an algebraic function. In this article, we will learn its definition, graph, various properties, etc. We will also learn how to find its derivative and integration.

Table of Contents

Cube Root Definition

If A³ =x, then the number A is called a cube root of x. Symbolically, we can write it as

Cube root Symbol:

Note: The cube root of x can be written as x^1/3, that is, = x^1/3.

Cube Root Example

As 2³ =8, by the above definition, we can say that 2 is a cube root of 8. More examples of cube roots are given below:

As 3³=27 , the number 3 is a cube root of 27.
As 4³ =64, the number 4 is a cube root of 64.
As 5³ =125, the number 5 is a cube root of 125.

Cube Root Graph

Let $y=$

Taking cubes on both sides, we get that

y³ =x.

The graph of cube root is plotted below:

Source: Wikipedia

Cube Root Properties

Below are the properties of cube roots:

The product of the cube roots of two numbers is the same as the cube root of the product of those two numbers. That is,
If $a, b (\neq 0)$ be two real numbers. Then we have
Cube root of a perfect cube is always an integer. Perfect cube is such a number that is the cube of some integer. For example, the cube root of $8$ is $2$, so $8$ is a perfect cube number.
The cube root of 0 is 0.
The cube roots of 1 are 1, ω, ω², where ω=$\dfrac{-1 \pm \sqrt{3}i}{2}$.
The cube root function is an odd function.

Cube Root Derivative

To find the derivative of cube root x, we will use the power rule: $\dfrac{d}{dx}(x^n)=nx^{n-1}$. We know that cube root of $x$ can be written as $x^{1/3}$.

Thus, by the above power rule, the derivative of

will be equal to

$\dfrac{d}{dx}(x^{1/3})$

$=\dfrac{1}{3} x^{1/3-1}$

$=\dfrac{1}{3} x^{-2/3}$

So the derivative of cube root x is 1/3 x^{-2/3}.

Also Read:

Derivative of root(x)

Derivative of 1/root(x)

Derivative of x^3

Derivative of x root(x)