$\"\"$

The function is $\"\"$ .

Differentiate on each side with respect to $\"\"$ .

$\"\"$

Apply power rule in derivatives $\"\"$ .

Apply chain rule in derivatives.

$\"\"$

$\"\"$ on $\"\"$ .

The derivative is always positive on $\"\"$ therefore $\"\"$ is increasing and thus has an inverse on this interval.

$\"\"$ is positive on the domain of $\"\"$ .

$\"\"$ is strictly monotonic and it must have inverse function.

$\"\"$

$\"\"$ on $\"\"$ .