\[ R'(x) = \frac{d}{dx}[\ln(x^2 + 4x + 5)] = \frac{2x + 4}{x^2 + 4x + 5} \] - Verified Servers

["Title: Understanding the Derivative of ln(x² + 4x + 5): Step-by-Step Explanation", "Meta Description:
\nLearn how to compute the derivative of \( \ln(x^2 + 4x + 5) \) with a clear, step-by-step explanation. We derive \( R'(x) = \frac{2x + 4}{x^2 + 4x + 5} \) using the chain rule, making it easier to understand this important differentiation rule.", "---", "### Introduction", "Calculating the derivative of a logarithmic function often involves a powerful technique known as the chain rule, especially when the argument is a more complex expression like a quadratic. In this article, we explore the exact derivative:", "\[
\nR'(x) = \frac{d}{dx} \left[ \ln(x^2 + 4x + 5) \right] = \frac{2x + 4}{x^2 + 4x + 5}
\n\]", "We’ll walk through the derivation methodically to build your confidence in differentiating logarithmic functions with polynomial arguments. Whether you're studying calculus or solving real-world optimization problems, mastering this concept is essential.", "---", "### The Chain Rule: Core Concept", "The chain rule states that if you have a composite function of the form \( \ln(u(x)) \), then its derivative is:", "\[
\n\frac{d}{dx}[\ln(u(x))] = \frac{u'(x)}{u(x)}
\n\]", "Here, \( u(x) = x^2 + 4x + 5 \), and we know that the derivative of \( \ln(u) \) is \( \frac{u'}{u} \). This is the foundation for computing derivatives of logarithms with non-constant arguments.", "---", "### Step-by-Step Derivation", "Let \( R(x) = \ln(x^2 + 4x + 5) \).", "1. Identify the inner function:
\nLet \( u(x) = x^2 + 4x + 5 \).
\nThen \( R(x) = \ln(u(x)) \).", "2. Differentiate the outer natural log function:
\n\[
\n\frac{d}{dx}[\ln(u(x))] = \frac{u'(x)}{u(x)}
\n\]", "3. Compute \( u'(x) \):
\nSince \( u(x) = x^2 + 4x + 5 \),
\n\[
\nu'(x) = 2x + 4
\n\]", "4. Substitute into the chain rule formula:
\n\[
\nR'(x) = \frac{u'(x)}{u(x)} = \frac{2x + 4}{x^2 + 4x + 5}
\n\]", "And there we have it — the derivative is fully derived.", "---", "### Why This Formula Matters", "This result demonstrates a widely used pattern in calculus:

\n

To differentiate \( \ln(f(x)) \), differentiate the inside function and divide by the function itself.", "Knowing this identity helps accurately solve derivatives involving logarithmic functions that model growth, probabilities, and other real-life phenomena.", "---", "### Tips for Applying the Formula", "- Always identify the inner function before differentiating.
\n- Use simple algebra to compute the derivative of the polynomial.
\n- Remember the chain rule applies no matter how complex \( u(x) \) is—polynomials included!
\n- Check your answer by differentiating \( \ln(x^2 + 4x + 5) \) using numerical values.", "---", "### Conclusion", "The derivative
\n\[
\nR'(x) = \frac{2x + 4}{x^2 + 4x + 5}
\n\]
\nis a textbook example of the chain rule applied to logarithmic functions. By breaking the process into clear steps—identify, differentiate the inside, and compose—the computation becomes intuitive. This principle unlocks more advanced calculus topics and practical problem-solving.", "Explore further derivatives of logs with quadratics and beyond—you’re now ready to tackle differentiation with confidence!", "---", "Keywords:
\nderivative of \(\ln(x^2 + 4x + 5)\), R'(x) = d/dx[ln(x² + 4x + 5)], chain rule derivative, calculus tutorial, polynomial logarithmic derivative, differentiation formula, math education.", "---", "Related Reads:
\n- How to differentiate natural logarithm functions
\n- Chain rule for composite functions explained
\n- Derivatives of logarithmic expressions with quadratics
\n- Practical examples of using ln(f(x)) in calculus problems", "---", "Mastering derivatives like this one builds a strong foundation for advanced math development—start practicing today!"]

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