https://file.aiquickdraw.com/mathbot/file/1728362897558xfi58r4e.jpg

The image contains two functions, \( f(x) = \ln(\sin^2(x)) \) and \( y = \frac{1}{\ln(x)} \), for which you are asking about the derivatives. I'll guide you through the process of finding these derivatives.

### 1. Derivative of \( f(x) = \ln(\sin^2(x)) \)
We will use the chain rule and logarithmic properties here.

First, apply the logarithmic identity:
\[
f(x) = 2 \ln(\sin(x))
\]

Now, differentiate using the chain rule:
\[
f'(x) = 2 \cdot \frac{1}{\sin(x)} \cdot \cos(x)
\]

Simplify:
\[
f'(x) = 2 \cdot \cot(x)
\]
Thus, the derivative of \( f(x) = \ln(\sin^2(x)) \) is:
\[
f'(x) = 2 \cot(x)
\]

### 2. Derivative of \( y = \frac{1}{\ln(x)} \)

This is a quotient, so we apply the quotient rule. Let:
\[
y = \frac{1}{\ln(x)} = (\ln(x))^{-1}
\]

The derivative of \( (\ln(x))^{-1} \) using the chain rule is:
\[
y' = -(\ln(x))^{-2} \cdot \frac{1}{x}
\]

Thus, the derivative of \( y = \frac{1}{\ln(x)} \) is:
\[
y' = -\frac{1}{x (\ln(x))^2}
\]

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Would you like further details on any step?

Here are 5 related questions you might consider:
1. How do we apply the product rule and chain rule together?
2. What is the derivative of \( \ln(x^2 + 1) \)?
3. How does the cotangent function behave in differentiation?
4. Can we simplify derivatives of logarithmic functions further in some cases?
5. What are some common mistakes when applying the chain rule in logarithmic functions?

**Tip**: When differentiating logarithmic functions, simplify them first using logarithmic properties like the power rule \( \ln(a^b) = b \ln(a) \), as it makes the differentiation easier.

What are the derivatives of f(x) = ln(sin^2(x)) and y = 1 / ln(x)?

Learn how to find the derivatives of f(x) = ln(sin^2(x)) and y = 1 / ln(x) using the chain rule and quotient rule. This problem covers key concepts in differentiation, logarithmic functions, and trigonometric functions. Suitable for university-level students studying Calculus.

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