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If for x ∈ (0, 1/4), the derivative of tan⁻¹((6x√x)/(1 − 9x³)) is √x · g(x), then g(x) equals:

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

Correct answer: 9/(1 + 9x³)

Solution

The correct option is derived from applying the chain rule and quotient rule to differentiate the given function, leading to the simplification that results in g(x) being expressed as 9/(1 + 9x³), which matches the form required by the problem.

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