Let's differentiate the function step by step. The given function is:

$y = \ln \left( \sin \left( \sqrt{\arctan \left( e^{2x} \right)} \right) \right)$

Step-by-step differentiation:

1. Outer function: The outermost function is the natural logarithm $\ln(u)$, where $u = \sin \left( \sqrt{\arctan \left( e^{2x} \right)} \right)$.

   • The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$.

2. First inner function: The next function is $\sin(v)$, where $v = \sqrt{\arctan \left( e^{2x} \right)}$.

   • The derivative of $\sin(v)$ with respect to $v$ is $\cos(v)$.

3. Second inner function: The next function is the square root $\sqrt{w}$, where $w = \arctan \left( e^{2x} \right)$.

   • The derivative of $\sqrt{w}$ with respect to $w$ is $\frac{1}{2\sqrt{w}}$.

4. Third inner function: The next function is $\arctan(z)$, where $z = e^{2x}$.

   • The derivative of $\arctan(z)$ with respect to $z$ is $\frac{1}{1+z^2}$.

5. Innermost function: The innermost function is $e^{2x}$.

   • The derivative of $e^{2x}$ with respect to $x$ is $2e^{2x}$.

Putting it all together:

By applying the chain rule, we differentiate each function with respect to its argument and multiply them together:

$\frac{dy}{dx} = \frac{1}{\sin \left( \sqrt{\arctan \left( e^{2x} \right)} \right)} \cdot \cos \left( \sqrt{\arctan \left( e^{2x} \right)} \right) \cdot \frac{1}{2 \sqrt{\arctan \left( e^{2x} \right)}} \cdot \frac{1}{1 + \left( e^{2x} \right)^2} \cdot 2e^{2x}$

This is the derivative of the given function.

Would you like further details or clarification on any part of the differentiation? Here's a list of related questions to explore:

1. How do you apply the chain rule to functions involving nested functions?
2. What is the derivative of inverse trigonometric functions like arctan?
3. How can you simplify complex derivatives step by step?
4. Can you differentiate logarithmic functions involving trigonometric expressions?
5. What are the common mistakes when differentiating nested functions?

Tip: When differentiating complex expressions, always carefully identify each nested function and apply the chain rule step by step.