Given xy+2x+3x2=9: 
(a) Find y′ by implicit differentiation (leave your answer in terms of x and y). 
(b) Solve the equation for y and differentiate to get y′ in terms of x. 
(The answers should be consistent!)

(a) y′=  
(b) y′=

To solve this problem, we will proceed with the following steps for parts (a) and (b) of the question:

### Given Equation:
\[
xy + 2x + 3x^2 = 9
\]

### Part (a): Implicit Differentiation
1. Differentiate both sides of the equation with respect to \( x \). Recall that \( y \) is a function of \( x \), so when differentiating \( y \), we must apply the chain rule and include \( y' \).
   
   Differentiating term by term:
   \[
   \frac{d}{dx}(xy) + \frac{d}{dx}(2x) + \frac{d}{dx}(3x^2) = \frac{d}{dx}(9)
   \]
   
2. Apply the product rule to the term \( xy \):
   \[
   \frac{d}{dx}(xy) = x \cdot \frac{dy}{dx} + y \cdot \frac{dx}{dx} = x y' + y
   \]
   
3. Differentiate the remaining terms:
   \[
   \frac{d}{dx}(2x) = 2 \quad \text{and} \quad \frac{d}{dx}(3x^2) = 6x
   \]

4. Substitute these derivatives back into the differentiated equation:
   \[
   x y' + y + 2 + 6x = 0
   \]

5. Solve for \( y' \):
   \[
   x y' = -y - 2 - 6x
   \]
   \[
   y' = \frac{-y - 2 - 6x}{x}
   \]

So, the answer for part (a) is:
\[
y' = \frac{-y - 2 - 6x}{x}
\]

### Part (b): Solve for \( y \) and Differentiate
1. Rearrange the original equation to solve for \( y \):
   \[
   xy + 2x + 3x^2 = 9
   \]
   \[
   xy = 9 - 2x - 3x^2
   \]
   \[
   y = \frac{9 - 2x - 3x^2}{x}
   \]

2. Differentiate \( y \) with respect to \( x \):
   \[
   y' = \frac{d}{dx} \left( \frac{9 - 2x - 3x^2}{x} \right)
   \]

3. Use the quotient rule, where \( u = 9 - 2x - 3x^2 \) and \( v = x \):
   \[
   y' = \frac{v \cdot u' - u \cdot v'}{v^2}
   \]

4. Compute \( u' \) and \( v' \):
   \[
   u' = -2 - 6x \quad \text{and} \quad v' = 1
   \]

5. Substitute these into the quotient rule:
   \[
   y' = \frac{x(-2 - 6x) - (9 - 2x - 3x^2)(1)}{x^2}
   \]

6. Simplify the expression:
   \[
   y' = \frac{-2x - 6x^2 - 9 + 2x + 3x^2}{x^2}
   \]
   \[
   y' = \frac{-9 - 3x^2}{x^2}
   \]

Both parts should yield consistent results in terms of structure.

Would you like further details on any steps? 

Here are some related questions:
1. How do we use implicit differentiation in other functions?
2. What does consistency between both methods indicate?
3. Why is the quotient rule useful in finding derivatives?
4. How does implicit differentiation handle terms with multiple variables?
5. Can we differentiate functions directly after simplifying them?

**Tip**: Consistency between two differentiation methods provides confidence in accuracy!

Given xy + 2x + 3x^2 = 9: (a) Find y′ by implicit differentiation (leave your answer in terms of x and y). (b) Solve the equation for y and differentiate to get y′ in terms of x. (The answers should be consistent!)

Explore a step-by-step solution involving implicit differentiation and the quotient rule to find derivatives for the equation xy + 2x + 3x² = 9. This detailed solution covers product and chain rules, providing consistent results through two methods of differentiation. Suitable for high school students in Grades 11-12.

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