Here are the step-by-step derivatives for questions 1, 3, and 7:

1. $y = \ln(8x)$

• The derivative of $\ln(u)$ is $\frac{1}{u} \cdot \frac{du}{dx}$.
• Let $u = 8x$, so $\frac{du}{dx} = 8$.
• The derivative of $y$ becomes: $\frac{dy}{dx} = \frac{1}{8x} \cdot 8 = \frac{1}{x}$

3. $y = \ln(8 - 3x)$

• Using the same rule for logarithmic differentiation:
• Let $u = 8 - 3x$, so $\frac{du}{dx} = -3$.
• The derivative of $y$ becomes: $\frac{dy}{dx} = \frac{1}{8 - 3x} \cdot (-3) = \frac{-3}{8 - 3x}$

7. $y = \ln(\sqrt{x + 5})$

• We rewrite $\ln(\sqrt{x + 5})$ as $\frac{1}{2} \ln(x + 5)$.
• Then, $\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{x + 5} = \frac{1}{2(x + 5)}$.

Let me know if you'd like further clarification or details!

Here are 5 additional questions related to these:

1. How does the chain rule apply to logarithmic differentiation?
2. What is the derivative of $\ln(x^2 + 1)$?
3. How do you find the derivative of a product involving logarithmic functions?
4. Can you apply implicit differentiation to logarithmic functions?
5. What is the relationship between natural logarithms and exponential functions when differentiating?

Tip: For complex logarithmic functions, always simplify the argument (inside the log) before differentiating for easier results.