`d/(dx)[tan^(-1)((sqrt(x)(3-x))/(1-3x))]=` (a) `1/(2(1+x)sqrt(x))` (b) `3/((1+x)sqrt(x))` `2/((1+x)sqrt(x))` (d) `3/(2(1+x)sqrt(x))`

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 solve the problem of differentiating \( \tan^{-1} \left( \frac{\sqrt{x}(3-x)}{1-3x} \right) \), we will follow these steps:

### Step 1: Simplify the expression
We start by substituting \( \sqrt{x} = \tan(\theta) \), which implies \( x = \tan^2(\theta) \). Thus, we can rewrite the expression as:
\[ \tan^{-1} \left( \frac{\tan(\theta)(3 - \tan^2(\theta))}{1 - 3\tan^2(\theta)} \right) \]

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