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For x in the interval (0, 1/√3), if the derivative of tan⁻¹ ((6x√(x))/(1-9x³)) can be written in the form √(x) g(x), then what is g(x)?

1. 3/(1+9x³)
2. 9/(1+9x³)
3. 3x√x/(1−9x³)
4. 3x/(1−9x³)

Correct answer: 9/(1+9x³)

Solution

The derivative of the function involves applying the chain rule and the derivative of the arctangent function, leading to a simplification that reveals the factor of sqrt{x} and the function g(x) as 9/(1+9x³), which matches the required form.

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