We are tasked with differentiating two functions:

A) $y = (\sin x)^{1 + x^2}$

B) $y = \ln \left( \frac{x^2 + 1}{x - 1} \right)$

I will solve each part step-by-step:

Part A: Differentiating $y = (\sin x)^{1 + x^2}$

We use the logarithmic differentiation technique because the power includes a variable expression.

1. Take the natural logarithm of both sides: $\ln y = (1 + x^2) \ln (\sin x)$

2. Differentiate both sides with respect to $x$, applying the product rule on the right-hand side: $\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left[ (1 + x^2) \ln (\sin x) \right]$

   Using the product rule: [ \frac{d}{dx} \left[ (1 + x^2) \ln (\sin x) \right] = \frac{d}{dx} (1 + x^2) \cdot \ln (\sin x) + (1 + x^2) \cdot \frac{d}{dx} [\ln (\sin x)] ]

   • $\frac{d}{dx} (1 + x^2) = 2x$
   • $\frac{d}{dx} [\ln (\sin x)] = \frac{1}{\sin x} \cdot \cos x = \cot x$

   So: $\frac{1}{y} \frac{dy}{dx} = 2x \ln (\sin x) + (1 + x^2) \cot x$

3. Multiply through by $y = (\sin x)^{1 + x^2}$ to isolate $\frac{dy}{dx}$: $\frac{dy}{dx} = (\sin x)^{1 + x^2} \left[ 2x \ln (\sin x) + (1 + x^2) \cot x \right]$

Part B: Differentiating $y = \ln \left( \frac{x^2 + 1}{x - 1} \right)$

Here we use the chain rule and the logarithmic property $\ln \left( \frac{u}{v} \right) = \ln u - \ln v$.

1. Split the logarithm: $y = \ln (x^2 + 1) - \ln (x - 1)$

2. Differentiate both terms separately:

   • For $\ln (x^2 + 1)$, use the chain rule: $\frac{d}{dx} \left[ \ln (x^2 + 1) \right] = \frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1}$

   • For $\ln (x - 1)$: $\frac{d}{dx} \left[ \ln (x - 1) \right] = \frac{1}{x - 1}$

3. Combine the results: $\frac{dy}{dx} = \frac{2x}{x^2 + 1} - \frac{1}{x - 1}$

Final Answers:

1. A) $\frac{dy}{dx} = (\sin x)^{1 + x^2} \left[ 2x \ln (\sin x) + (1 + x^2) \cot x \right]$
2. B) $\frac{dy}{dx} = \frac{2x}{x^2 + 1} - \frac{1}{x - 1}$

Let me know if you'd like a detailed verification for any step or further clarification!

Related Questions:

1. How do you differentiate functions with variable exponents using logarithmic differentiation?
2. What is the product rule, and how does it apply to differentiation?
3. Can you explain how to differentiate logarithmic functions like $\ln \left( \frac{u}{v} \right)$?
4. How does the chain rule work in composite functions such as ( \ln (\sin x