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To solve for \( y' \) using logarithmic differentiation, let's go through the following steps:

Given:
\[
y = \frac{\sin^2(x) \cdot \tan^6(x)}{(x^2 + 3)^2}
\]

### Step 1: Take the Natural Logarithm of Both Sides
Applying the natural logarithm to both sides, we get:
\[
\ln(y) = \ln\left(\frac{\sin^2(x) \cdot \tan^6(x)}{(x^2 + 3)^2}\right)
\]

Using logarithmic properties (specifically, \(\ln \frac{a}{b} = \ln a - \ln b\) and \(\ln(ab) = \ln a + \ln b\)), this becomes:
\[
\ln(y) = \ln(\sin^2(x)) + \ln(\tan^6(x)) - \ln((x^2 + 3)^2)
\]

Now we can simplify further by bringing down the exponents:
\[
\ln(y) = 2 \ln(\sin(x)) + 6 \ln(\tan(x)) - 2 \ln(x^2 + 3)
\]

### Step 2: Differentiate Both Sides with Respect to \( x \)
Now, differentiate implicitly with respect to \( x \):

\[
\frac{1}{y} \cdot y' = 2 \cdot \frac{\cos(x)}{\sin(x)} + 6 \cdot \frac{1}{\tan(x)} \cdot \sec^2(x) - 2 \cdot \frac{2x}{x^2 + 3}
\]

This simplifies to:
\[
\frac{y'}{y} = 2 \cot(x) + 6 \sec^2(x) - \frac{4x}{x^2 + 3}
\]

### Step 3: Solve for \( y' \)
Now, multiply both sides by \( y \) to isolate \( y' \):
\[
y' = y \cdot \left(2 \cot(x) + 6 \sec^2(x) - \frac{4x}{x^2 + 3}\right)
\]

Finally, substitute back the expression for \( y \):
\[
y' = \frac{\sin^2(x) \cdot \tan^6(x)}{(x^2 + 3)^2} \cdot \left(2 \cot(x) + 6 \sec^2(x) - \frac{4x}{x^2 + 3}\right)
\]

This is the derivative \( y' \) of the function.

Would you like more details on any part of this process or have any questions?

Here are five related questions for further practice:
1. How would the process change if the exponent of \(\sin(x)\) was 3 instead of 2?
2. Can you use logarithmic differentiation on a product involving exponential and trigonometric functions?
3. How do you simplify expressions involving \(\cot(x)\) and \(\sec^2(x)\) after differentiation?
4. What if we applied logarithmic differentiation to only the numerator? Would that be valid?
5. How does the derivative of a function change if it’s under a square root?

**Tip:** When using logarithmic differentiation, always simplify by expanding the log expression first before differentiating—it makes the process more straightforward.

Use logarithmic differentiation to find the derivative of the function: y = (sin^2(x) * tan^6(x)) / (x^2 + 3)^2.

Learn how to use logarithmic differentiation to find the derivative of the function y = (sin^2(x) * tan^6(x)) / (x^2 + 3)^2. This problem demonstrates the application of logarithmic differentiation, implicit differentiation, and trigonometric derivatives. Suitable for advanced high school calculus students.

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