`d/(dx)[tan^(-1)((sqrt(x)(3-x))/(1-3x))]=`

Options

A. `(1)/(2(1+x)sqrt(x))`
B. `(3)/((1+x)sqrt(x))`
C. `(2)/((1+x)sqrt(x))`
D. `(3)/(2(1+x)sqrt(x))`

Explanation

To find the derivative of the function \( \tan^{-1} \left( \frac{\sqrt{x}(3-x)}{1-3x} \right) \), we will follow these steps:

### Step 1: Rewrite the function
Let \( y = \tan^{-1} \left( \frac{\sqrt{x}(3-x)}{1-3x} \right) \).

### Step 2: Use the substitution
We will use the substitution \( \sqrt{x} = \tan(\theta) \). Therefore, \( x = \tan^2(\theta) \) and \( \frac{d}{dx} = \frac{d\theta}{dx} \cdot \frac{d}{d\theta} \).

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