Consider the function:f(x)=ln(((1x+2)^5)3√(x^4+2))/((2x+5)^3). Find f′(0).

To find \( f'(0) \), we will first differentiate the function \( f(x) = \ln \left( \frac{((x+2)^5) \sqrt[3]{x^4+2}}{(2x+5)^3} \right) \) with respect to \( x \), and then substitute \( x = 0 \) into the result.

### Step 1: Rewrite the function using properties of logarithms

Using the property \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \), we can rewrite \( f(x) \) as:
\[
f(x) = \ln \left( (x+2)^5 \cdot \sqrt[3]{x^4+2} \right) - \ln \left( (2x+5)^3 \right).
\]

Now, applying the properties \( \ln(a \cdot b) = \ln(a) + \ln(b) \), \( \ln(a^n) = n \ln(a) \), we get:
\[
f(x) = 5 \ln(x+2) + \frac{1}{3} \ln(x^4 + 2) - 3 \ln(2x+5).
\]

### Step 2: Differentiate \( f(x) \) with respect to \( x \)

Now we differentiate each term in \( f(x) \) separately.

1. **First Term: \( 5 \ln(x+2) \)**
   \[
   \frac{d}{dx} \left( 5 \ln(x+2) \right) = \frac{5}{x+2}.
   \]

2. **Second Term: \( \frac{1}{3} \ln(x^4 + 2) \)**
   \[
   \frac{d}{dx} \left( \frac{1}{3} \ln(x^4 + 2) \right) = \frac{1}{3} \cdot \frac{4x^3}{x^4 + 2} = \frac{4x^3}{3(x^4 + 2)}.
   \]

3. **Third Term: \( -3 \ln(2x+5) \)**
   \[
   \frac{d}{dx} \left( -3 \ln(2x+5) \right) = -3 \cdot \frac{2}{2x+5} = -\frac{6}{2x+5}.
   \]

Putting it all together, we have:
\[
f'(x) = \frac{5}{x+2} + \frac{4x^3}{3(x^4 + 2)} - \frac{6}{2x+5}.
\]

### Step 3: Substitute \( x = 0 \) into \( f'(x) \)

Now, substitute \( x = 0 \):
\[
f'(0) = \frac{5}{0+2} + \frac{4 \cdot 0^3}{3(0^4 + 2)} - \frac{6}{2 \cdot 0 + 5}.
\]

Simplifying each term:
1. \( \frac{5}{2} = 2.5 \)
2. \( \frac{4 \cdot 0}{3 \cdot 2} = 0 \)
3. \( -\frac{6}{5} = -1.2 \)

Thus,
\[
f'(0) = 2.5 + 0 - 1.2 = 1.3.
\]

### Final Answer
\[
f'(0) = 1.3.
\]

Would you like additional details, or have any questions?

Here are some related questions to explore further:
1. How would the differentiation change if \(f(x)\) contained an additional nested function?
2. How do properties of logarithms simplify the differentiation process?
3. What other functions might require similar techniques (logarithmic differentiation)?
4. How can chain rule applications be expanded for complex functions like this?
5. How does substituting at \(x = 0\) change results compared to other points?

**Tip:** Remember to check for simplification opportunities using logarithmic properties in complex functions for easier differentiation.

Consider the function: f(x) = ln(((x+2)^5) * 3√(x^4+2) / (2x+5)^3). Find f′(0).

Find the derivative f′(0) for the function f(x) = ln(((x+2)^5) * 3√(x^4+2) / (2x+5)^3) using logarithmic differentiation and properties of logarithms. This solution demonstrates step-by-step differentiation techniques suitable for advanced calculus students.

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