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Can you do questions 1,3,7 and show step by step

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.

Find the derivative of each function for the functions y = ln(8x), y = ln(8 - 3x), and y = ln(sqrt(x + 5)).

Explore detailed steps in logarithmic differentiation for functions ln(8x), ln(8 - 3x), and ln(sqrt(x + 5)). Understand how to apply the chain rule in calculus to solve these problems effectively.

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