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The task is to find the derivative of the function \( f(x) = \log_2(2x^2 - 3x + 7) \).

### Steps:

1. **Rewrite the function**:
   \[
   f(x) = \log_2(u), \text{ where } u = 2x^2 - 3x + 7.
   \]

2. **Apply the chain rule**:
   The derivative of \( \log_b(x) \) is \( \frac{1}{x \ln(b)} \), where \( b \) is the base of the logarithm.
   
   Using this, the derivative of \( f(x) = \log_2(u) \) is:
   \[
   f'(x) = \frac{1}{u \ln(2)} \cdot \frac{du}{dx}.
   \]

3. **Find the derivative of \( u \)**:
   \[
   u = 2x^2 - 3x + 7,
   \]
   \[
   \frac{du}{dx} = 4x - 3.
   \]

4. **Substitute everything back**:
   \[
   f'(x) = \frac{4x - 3}{(2x^2 - 3x + 7) \ln(2)}.
   \]

Thus, the derivative of the function is:
\[
f'(x) = \frac{4x - 3}{(2x^2 - 3x + 7) \ln(2)}.
\]

Would you like more details on any part of this process? 

Here are 5 related questions:

1. How do you compute the derivative of logarithmic functions with different bases?
2. How does the chain rule apply to composite functions like \( \log_2(2x^2 - 3x + 7) \)?
3. Can the logarithmic differentiation method be applied here? How would it work?
4. What are some real-world applications of logarithmic functions and their derivatives?
5. How do changes in the base of the logarithm affect the differentiation?

**Tip:** Always remember that when differentiating logarithms with bases other than \(e\), a factor of \( \ln(b) \) appears in the denominator.

Apskaičiuokite funkcijos f(x) išvestinę, kai f(x) = log_2(2x^2 - 3x + 7).

Learn how to compute the derivative of the logarithmic function f(x) = log_2(2x^2 - 3x + 7) using the chain rule. This detailed solution includes key calculus concepts like logarithmic differentiation. Suitable for students in Grades 11-12.

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