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

We are tasked with finding the derivative of \( y = \ln \sqrt{\frac{2x+1}{x^2-4}} \) at \( x = 4 \).

### Step 1: Simplify the Expression
We can rewrite the logarithmic expression using the logarithm properties:
\[
y = \ln \sqrt{\frac{2x+1}{x^2-4}} = \frac{1}{2} \ln \left( \frac{2x+1}{x^2-4} \right)
\]

### Step 2: Differentiate
Now, let's differentiate \( y = \frac{1}{2} \ln \left( \frac{2x+1}{x^2-4} \right) \) with respect to \( x \) using the chain rule and quotient rule. The derivative of \( \ln(f(x)) \) is \( \frac{f'(x)}{f(x)} \), so we proceed by applying that.

Let:
\[
 f(x) = \frac{2x + 1}{x^2 - 4}
\]

Now differentiate \( f(x) \) using the quotient rule:
\[
f'(x) = \frac{(2x^2 - 4x) - (2x + 1)(2x)}{(x^2 - 4)^2}
= \frac{2x^2 - 4x - (4x^2 + 2x)}{(x^2 - 4)^2}
= \frac{-2x^2 - 6x}{(x^2 - 4)^2}
= \frac{-2x(x + 3)}{(x^2 - 4)^2}
\]

Now, the derivative of \( y \) is:
\[
\frac{dy}{dx} = \frac{1}{2} \cdot \frac{f'(x)}{f(x)} = \frac{1}{2} \cdot \frac{-2x(x + 3)}{(x^2 - 4)^2} \cdot \frac{x^2 - 4}{2x + 1}
= \frac{-x(x+3)(x^2-4)}{(2x+1)(x^2-4)^2}
\]
Simplifying:
\[
\frac{dy}{dx} = \frac{-x(x+3)}{(2x+1)(x^2-4)}
\]

### Step 3: Evaluate at \( x = 4 \)
Substitute \( x = 4 \) into the derivative expression:
\[
\frac{dy}{dx} = \frac{-4(4+3)}{(2(4)+1)((4)^2 - 4)} = \frac{-4(7)}{(8+1)(16-4)} = \frac{-28}{9 \cdot 12} = \frac{-28}{108} = \frac{-7}{27}
\]

### Final Answer:
The derivative of \( y \) at \( x = 4 \) is \( \frac{-7}{27} \).

Would you like further details or have any questions?

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Here are some related questions to explore:
1. How do you apply the quotient rule in differentiation?
2. What is the derivative of \( \ln x \) and how does it apply here?
3. How does the chain rule work with nested functions like logarithms?
4. What role does logarithmic differentiation play in simplifying derivatives?
5. How would you compute higher-order derivatives for this function?

**Tip:** Always simplify the expression before differentiation to make the process more manageable!

Learn how to find the derivative of y = ln(sqrt((2x+1)/(x^2-4))) at x = 4 using the quotient rule and chain rule. This problem involves logarithmic differentiation and applying key calculus principles. Suitable for students in Grades 11-12.

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