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## Definitions

### General

The cube root of a number x, is a number r so that r^{3} = x. In modern notation cube root is written as \(\sqrt[3]{x}\) or x^{1/3} . Any real number has one real cube root, while any non-zero real number has two complex cube roots too. The real cube root of a positive number is always positive while the real cube root of a negative number is negative. The cube root of 0 is 0.

The graph of the real cube root function is shown in the figure below. It is a monotonic function with symmetry around origin.

### Properties

The derivative of the cube root function is:

\[ \left(\sqrt[3] x\right)' = \frac{1}{3}x^{-2/3} = \frac{1}{3\sqrt[3]{x^2}} \]

The integral of the cube root function is given by:

\[ \int \sqrt[3] x\, \mathrm{d}x = \frac{3x^{4/3}}{4} +C \]