To find \(\frac{dy}{dx}\) for \(y = \ln(x^2 + x + 4)\), we can apply the chain rule. Here's how to do it:

1. First, recognize that \(y = \ln(f(x))\), where \(f(x) = x^2 + x + 4\).

2. The derivative of \(\ln(f(x))\) with respect to \(x\) is given by:

\[
\frac{dy}{dx} = \frac{1}{f(x)} \cdot \frac{d}{dx}(f(x))
\]

3. Now, find \(\frac{d}{dx}(f(x))\) for \(f(x) = x^2 + x + 4\):

\[
\frac{d}{dx}(x^2 + x + 4) = 2x + 1
\]

4. Substitute \(f(x)\) and its derivative into the formula:

\[
\frac{dy}{dx} = \frac{1}{x^2 + x + 4} \cdot (2x + 1)
\]

So, the derivative is:

\[
\frac{dy}{dx} = \frac{2x + 1}{x^2 + x + 4}
\]

Let me know if you'd like further explanation or details on any of the steps.

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**Related Questions:**
1. How would the derivative change if the function inside the logarithm was \(x^3 + x + 4\)?
2. What happens if you differentiate \(y = \ln(2x^2 + 3x + 5)\)?
3. How can the chain rule be used when differentiating logarithmic functions involving more complex terms?
4. What is the derivative of \(y = e^{x^2 + x + 4}\)?
5. How do logarithmic derivatives simplify certain problems involving products or quotients?

**Tip:** The chain rule is essential when differentiating composite functions like \(y = \ln(f(x))\). Always differentiate the outer function first, and then multiply by the derivative of the inner function.

Learn how to find the derivative of y = ln(x^2 + x + 4) using the chain rule. This solution demonstrates step-by-step how to differentiate composite logarithmic functions. Suitable for high school calculus students in Grades 11-12.

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