https://file.aiquickdraw.com/mathbot/file/1726502035844giy267gg.jpg

The expression provided in the image is:

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

To find the derivative of this expression with respect to \(x\), we'll follow these steps:

1. **Simplify the expression**: Using the logarithmic property \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\), we rewrite the expression as:
   \[
   \ln\left(\frac{1}{(x+1)^2}\right) = \ln(1) - \ln((x+1)^2)
   \]
   Since \(\ln(1) = 0\), this simplifies to:
   \[
   - \ln((x+1)^2)
   \]

2. **Use another logarithmic property**: Apply the logarithmic property \(\ln(a^b) = b \ln(a)\):
   \[
   - \ln((x+1)^2) = -2 \ln(x+1)
   \]

3. **Differentiate**: Now, differentiate \(-2 \ln(x+1)\) with respect to \(x\). The derivative of \(\ln(x+1)\) is \(\frac{1}{x+1}\), so:
   \[
   \frac{d}{dx} \left(-2 \ln(x+1)\right) = -2 \cdot \frac{1}{x+1}
   \]
   Simplifying this, we get:
   \[
   -\frac{2}{x+1}
   \]

Thus, the derivative of the given expression is:
\[
-\frac{2}{x+1}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you differentiate \(\ln\left(\frac{x+1}{x-1}\right)\)?
2. What happens if the base of the logarithm changes, e.g., \(\log_b\left(\frac{1}{(x+1)^2}\right)\)?
3. How would the result change if the exponent in the denominator was different, say \( (x+1)^3 \)?
4. Can you find the second derivative of the expression \(\ln\left(\frac{1}{(x+1)^2}\right)\)?
5. What is the general rule for differentiating logarithmic functions?

**Tip**: Always simplify logarithmic expressions using properties before differentiating, as it reduces complexity.

Learn how to differentiate the logarithmic expression ln(1/(x+1)^2) step by step using logarithmic properties and the chain rule. This solution simplifies the expression before applying the derivative.

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