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The given function is \( y = \sqrt{2 - 3x} \).

To find the derivative \( y' \) of this function, we'll use the chain rule. Here's the step-by-step process:

1. **Recognize the outer function and inner function:**
   - Outer function: \( f(u) = \sqrt{u} \) with derivative \( f'(u) = \frac{1}{2\sqrt{u}} \).
   - Inner function: \( g(x) = 2 - 3x \) with derivative \( g'(x) = -3 \).

2. **Apply the chain rule:**
   - The derivative of \( y \) with respect to \( x \) is given by \( f'(g(x)) \cdot g'(x) \).

3. **Substitute and simplify:**
   \[
   y' = \frac{1}{2\sqrt{2 - 3x}} \cdot (-3) = \frac{-3}{2\sqrt{2 - 3x}}
   \]

Thus, the derivative of \( y = \sqrt{2 - 3x} \) is \( y' = \frac{-3}{2\sqrt{2 - 3x}} \).

Would you like further details on this calculation or have any other questions? Here are some follow-up questions you might consider:

1. How does the domain of the original function affect its derivative?
2. What happens to the derivative as \( x \) approaches the boundaries of the domain?
3. Can we determine the critical points of the function from the derivative?
4. How does the concavity of the function change across its domain?
5. What real-world scenarios could be modeled by an equation like \( y = \sqrt{2 - 3x} \)?

**Tip:** When applying the chain rule, always clearly identify the outer and inner functions to avoid mistakes in your derivative calculation.

Find the derivative of the function y = √(2 - 3x)

Learn how to find the derivative of y = √(2 - 3x) by applying the chain rule. This detailed solution covers the outer and inner functions, key steps, and provides the final derivative as y' = (-3) / (2√(2 - 3x)). Suitable for students in Grades 11-12.

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