Thank you for sharing the image! I can analyze and work through the math problems on the board. Here are the equations written out for clarity:

1. $y = 2 \sin \left[ \ln \left( \cos \left( \sqrt{x^2 + 3x} \right) \right) \right]$
2. $y = 5 \ln \left[ \left( x^2 + 1 \right)^7 (3x) \right]$
3. $y = \sqrt[3]{5e^{x^2} + \cos^2 (4x^3)}$
4. $y = 7 \cot^2 \left( \sqrt{\sin \left( \tan \left( e^{2x} \right) \right)} \right)$

Would you like solutions for specific problems or all of them? Let me know if you'd like any particular kind of analysis (e.g., derivatives, domain checks, simplification, etc.).

For now, here are relative questions to deepen your understanding:

1. What is the general procedure to find the derivative of logarithmic and trigonometric functions?
2. How would you simplify $\ln(a^b \cdot c)$ for easier computation in question 2?
3. What conditions should $x$ satisfy for the expressions inside logarithmic and square root functions to remain valid?
4. How do you compute higher-order derivatives for nested trigonometric functions like in question 4?
5. What is the difference between solving these equations symbolically and numerically?

Tip: For complex compositions of functions, identifying the innermost function first helps streamline differentiation or integration tasks.