https://file.aiquickdraw.com/mathbot/file/1725243679820zd11yv32.jpg

The expression in the image is:

\[
f(x) = \log(3x^2 + 9x - 13)
\]

To differentiate this function with respect to \(x\), we'll apply the chain rule.

### Step-by-Step Differentiation:

1. **Outer Function**: The outer function is the logarithm, \(\log(u)\), where \(u = 3x^2 + 9x - 13\).

   The derivative of \(\log(u)\) with respect to \(u\) is:
   \[
   \frac{d}{du} \log(u) = \frac{1}{u}
   \]

2. **Inner Function**: The inner function is \(u = 3x^2 + 9x - 13\).

   The derivative of \(u\) with respect to \(x\) is:
   \[
   \frac{du}{dx} = \frac{d}{dx} (3x^2 + 9x - 13) = 6x + 9
   \]

3. **Applying the Chain Rule**: 
   \[
   \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = \frac{1}{3x^2 + 9x - 13} \cdot (6x + 9)
   \]

### Final Answer:
\[
f'(x) = \frac{6x + 9}{3x^2 + 9x - 13}
\]

Would you like further details or have any questions about this differentiation process?

### Related Questions:
1. What are the critical points of \( f(x) = \log(3x^2 + 9x - 13) \)?
2. How would you find the second derivative of \( f(x) \)?
3. What is the domain of \( f(x) \)?
4. How can you determine the concavity of \( f(x) \) using the second derivative?
5. Can you integrate \( f(x) \)? If so, what is the integral?

### Tip:
When differentiating functions involving logarithms, always remember to apply the chain rule, starting from the outermost function.

Learn how to differentiate the logarithmic function f(x) = log(3x^2 + 9x - 13) step-by-step. This problem involves applying the chain rule in calculus and understanding logarithmic differentiation. Suitable for advanced high school students and calculus learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation