differentiate y=ln(x^2 sin x / (1+x)^1/2 )

We are asked to differentiate the function:

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

### Step-by-Step Solution:

#### Step 1: Apply the logarithmic properties
Using the properties of logarithms, we can simplify the expression inside the logarithm:

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

Simplifying further:

\[
y = 2\ln(x) + \ln(\sin x) - \frac{1}{2} \ln(1+x)
\]

#### Step 2: Differentiate term-by-term
Now, we differentiate each term individually with respect to \(x\).

1. \( \frac{d}{dx} [2\ln(x)] = \frac{2}{x} \)
2. \( \frac{d}{dx} [\ln(\sin x)] = \frac{1}{\sin x} \cdot \cos x = \cot x \)
3. \( \frac{d}{dx} \left[ -\frac{1}{2} \ln(1+x) \right] = -\frac{1}{2} \cdot \frac{1}{1+x} = -\frac{1}{2(1+x)} \)

#### Step 3: Combine the results
Now, we combine all the differentiated terms:

\[
\frac{dy}{dx} = \frac{2}{x} + \cot x - \frac{1}{2(1+x)}
\]

Thus, the derivative of \( y \) is:

\[
\frac{dy}{dx} = \frac{2}{x} + \cot x - \frac{1}{2(1+x)}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How do you differentiate more complex logarithmic functions?
2. What are the rules for differentiating trigonometric functions?
3. How can the chain rule be applied to logarithmic derivatives?
4. What is the significance of logarithmic differentiation?
5. How do you handle differentiation of products and quotients involving logarithms?

**Tip:** Always look for ways to simplify complex functions using logarithmic properties before differentiating!

This detailed solution covers the differentiation of y = ln(x^2 sin x / sqrt(1+x)) using logarithmic properties, chain rule, product rule, and trigonometric differentiation. Ideal for college-level calculus students.

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