find the derivative of y= ln\sqrt((2x+1)/(x^(2)-4) )  at x=4

We are tasked with finding the derivative of the function:

\[
y = \ln \left( \sqrt{\frac{2x+1}{x^2 - 4}} \right)
\]

### Step 1: Simplify the expression
We can first simplify the logarithmic expression by using the logarithmic identity:

\[
\ln(\sqrt{u}) = \frac{1}{2} \ln(u)
\]

Thus, the function becomes:

\[
y = \frac{1}{2} \ln \left( \frac{2x+1}{x^2 - 4} \right)
\]

### Step 2: Differentiate using the chain rule
We now apply the chain rule and the properties of logarithms. The derivative of \( \ln \left( \frac{a}{b} \right) \) is:

\[
\frac{d}{dx} \ln \left( \frac{a}{b} \right) = \frac{1}{a} \frac{da}{dx} - \frac{1}{b} \frac{db}{dx}
\]

Let \( a = 2x + 1 \) and \( b = x^2 - 4 \).

Now differentiate \( a \) and \( b \) with respect to \( x \):

\[
\frac{da}{dx} = 2, \quad \frac{db}{dx} = 2x
\]

Using the product of the logarithmic derivative:

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

This simplifies to:

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

### Step 3: Evaluate at \( x = 4 \)
Substitute \( x = 4 \) into the derivative:

\[
\frac{dy}{dx} = \frac{1}{2} \left( \frac{2}{2(4) + 1} - \frac{2(4)}{4^2 - 4} \right)
\]

Simplify each term:

\[
\frac{dy}{dx} = \frac{1}{2} \left( \frac{2}{9} - \frac{8}{12} \right) = \frac{1}{2} \left( \frac{2}{9} - \frac{2}{3} \right)
\]

Find a common denominator:

\[
\frac{dy}{dx} = \frac{1}{2} \left( \frac{2}{9} - \frac{6}{9} \right) = \frac{1}{2} \left( \frac{-4}{9} \right)
\]

Thus:

\[
\frac{dy}{dx} = \frac{-4}{18} = \frac{-2}{9}
\]

### Final Answer:
The derivative of \( y \) at \( x = 4 \) is:

\[
\frac{dy}{dx} = -\frac{2}{9}
\]

Would you like more details or have any questions?

Here are five related questions to consider:
1. How do logarithmic properties simplify the process of differentiation?
2. Can the derivative of a square root be generalized for any function?
3. How does the chain rule apply when differentiating logarithmic functions?
4. What are common mistakes when applying the quotient rule in derivatives?
5. How do second derivatives work for logarithmic functions?

**Tip:** Simplifying expressions before differentiating often makes the process much easier!

Find the derivative of the function y = ln(sqrt((2x+1)/(x^2 - 4))) at x = 4. This problem involves logarithmic differentiation, the quotient rule, and the chain rule. Suitable for college-level calculus students. Step-by-step solution included.

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