Differentiate `tan^(-1)((1+a x)/(1-a x))` with respect to `sqrt(1+a^2x^2)dot`

Explanation

To differentiate the function \( u = \tan^{-1}\left(\frac{1 + ax}{1 - ax}\right) \) with respect to \( v = \sqrt{1 + a^2x^2} \), we will use the chain rule. We will find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \), and then compute \( \frac{du}{dv} = \frac{du/dx}{dv/dx} \).

### Step-by-Step Solution:

1. **Define the Functions**:
\[ u = \tan^{-1}\left(\frac{1 + ax}{1 - ax}\right), \quad v = \sqrt{1 + a^2x^2} \]
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