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.

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)