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The image you uploaded contains a series of mathematical problems involving derivatives, trigonometric functions, implicit differentiation, and chain rule applications. Here’s a breakdown of the problems and an approach to solving each:

1. \( y = \sin^2(3x - 4) \)  
   - Differentiate \( y \) with respect to \( x \), using the chain rule.

2. \( y = \frac{\cos 2x}{3x - 4} \)  
   - Use the quotient rule to differentiate this expression.

3. \( y = \frac{1}{2} \tan (x \sin 2x) \)  
   - Apply the chain rule to differentiate this function, considering both the tangent and sine functions.

4. Find \( y'' - 2y' \) when \( y = \sin 3x - \cos 3x \).  
   - First, compute the first and second derivatives of \( y \) and then substitute into the given expression.

5. \( xy + 2x - 5y - y^2 = 0 \)  
   - Use implicit differentiation to find \( \frac{dy}{dx} \).

6. \( y = \sin^2(x^2 - x) \)  
   - Differentiate using the chain rule.

7. Find \( \frac{dy}{dx} \) from the equation \( xy^2 + y - x^2 = 6 \).  
   - Again, use implicit differentiation to find \( \frac{dy}{dx} \).

Would you like a step-by-step solution to any of these problems? Let me know which one you'd like to start with, and I'll guide you through it!

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Here are 5 questions to expand on the information:
1. How do you apply the chain rule in problem 1 to find the derivative of \( \sin^2(3x - 4) \)?
2. What is the quotient rule and how is it applied in problem 2?
3. Can you differentiate a composite function like in problem 3? How would you use the chain rule?
4. What is implicit differentiation and how would you apply it to problem 5?
5. In problem 7, how do you differentiate both sides of an equation like \( xy^2 + y - x^2 = 6 \)?

**Tip:** When performing implicit differentiation, always remember to apply the product rule where necessary (for example, with terms like \( xy \)).

Derivatives and trigonometric functions problems: y = sin^2(3x - 4), y = (cos 2x) / (3x - 4), y = (1/2) * tan(x * sin(2x)), Find y'' - 2y' when y = sin(3x) - cos(3x), Solve xy + 2x - 5y - y^2 = 0, Differentiate y = sin^2(x^2 - x) using chain rule, Find dy/dx for xy^2 + y - x^2 = 6.

Explore detailed solutions to advanced derivative problems involving trigonometric functions, implicit differentiation, the chain rule, and the quotient rule. This guide covers finding y'' - 2y', implicit differentiation for equations like xy + 2x - 5y - y^2 = 0, and using the chain rule on expressions like y = sin^2(x^2 - x). Perfect for high school students in Grades 11-12.

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