https://file.aiquickdraw.com/mathbot/file/1728515418968lodwza62.jpg

We are asked to find the derivative of the following function:

\[
y = \ln\left(\sqrt{\frac{x^2 - 11}{x^2 + 11}}\right)
\]

Let's break it down step by step:

### Step 1: Simplify the expression inside the logarithm
We know that \(\ln(\sqrt{u}) = \frac{1}{2}\ln(u)\), where \(u\) is a function of \(x\). Applying this rule, we can rewrite the expression as:

\[
y = \frac{1}{2} \ln\left(\frac{x^2 - 11}{x^2 + 11}\right)
\]

### Step 2: Use the logarithmic property \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
This allows us to separate the terms:

\[
y = \frac{1}{2} \left[\ln(x^2 - 11) - \ln(x^2 + 11)\right]
\]

### Step 3: Differentiate using the chain rule
Now, we differentiate each part with respect to \(x\). Let's handle each term separately:

1. For \(\ln(x^2 - 11)\), the derivative is:
   \[
   \frac{d}{dx} \ln(x^2 - 11) = \frac{1}{x^2 - 11} \cdot \frac{d}{dx}(x^2 - 11) = \frac{2x}{x^2 - 11}
   \]

2. For \(\ln(x^2 + 11)\), the derivative is:
   \[
   \frac{d}{dx} \ln(x^2 + 11) = \frac{1}{x^2 + 11} \cdot \frac{d}{dx}(x^2 + 11) = \frac{2x}{x^2 + 11}
   \]

### Step 4: Combine the results
Now, putting it all together:

\[
\frac{dy}{dx} = \frac{1}{2} \left( \frac{2x}{x^2 - 11} - \frac{2x}{x^2 + 11} \right)
\]

### Step 5: Factor and simplify
We can factor out the common term \(2x\):

\[
\frac{dy}{dx} = \frac{2x}{2} \left( \frac{1}{x^2 - 11} - \frac{1}{x^2 + 11} \right)
\]

\[
\frac{dy}{dx} = x \left( \frac{1}{x^2 - 11} - \frac{1}{x^2 + 11} \right)
\]

### Step 6: Combine the fractions
We can combine the two fractions under a common denominator:

\[
\frac{dy}{dx} = x \left( \frac{(x^2 + 11) - (x^2 - 11)}{(x^2 - 11)(x^2 + 11)} \right)
\]

Simplifying the numerator:

\[
\frac{dy}{dx} = x \left( \frac{x^2 + 11 - x^2 + 11}{(x^2 - 11)(x^2 + 11)} \right) = x \left( \frac{22}{(x^2 - 11)(x^2 + 11)} \right)
\]

### Final Answer:
\[
\frac{dy}{dx} = \frac{22x}{(x^2 - 11)(x^2 + 11)}
\]

This is the derivative of the given function.

Would you like further details or have any questions?

### 5 Related Questions:
1. How would the result change if the logarithmic function was natural log squared?
2. Can we find the second derivative of this function?
3. What is the domain of the original function?
4. How does this process change for logarithms with a different base (e.g., log base 10)?
5. How does simplifying the function before differentiation help in solving the derivative?

### Tip:
When differentiating complex expressions, breaking them down using logarithmic properties and simplifying before applying the chain rule can save a lot of time and reduce errors.

Find the derivative of y = ln(sqrt((x^2 - 11)/(x^2 + 11))).

Learn how to find the derivative of y = ln(sqrt((x^2 - 11)/(x^2 + 11))). This problem involves logarithmic differentiation, the chain rule, and simplifying complex logarithmic expressions. Detailed steps include breaking down the logarithm using the quotient rule and applying the chain rule for advanced calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation