Derivative of root functions

Root functions are differentiated by first converting them to a fractional power. The familiar power derivation rules is then used.

The fractional power functions are continuous and their domain is

Example 1

Example 2

Example 3

Differentiate the function

Note that this is a combined function. We must remember the derivative of the inner function.

Note. The domain of the derivative function is the open interval -1 < x < 1

Example 4

Differentiate the function

Domain of definition

Product rule of derivative

Example 5

Find the minimum value of the function in the previous example

The continuous function gets its minimum value at the zero point of the derivative or at the end point of the interval

Zeros of the derivative function is found by zeros of the numerator

The zero point belongs to the function domain. We create a behavior chart of the function

Based on the behavior chart, the minimum value is obtained in the zero of the derivative

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