https://file.aiquickdraw.com/mathbot/file/17274073663499pxg1359.jpg

Let's work through the problems shown in the image.

### Problem 15:  
\[ y = \frac{1}{3x^2 - 5} \]

To find the derivative \( y' \), we will apply the quotient rule or consider it as a chain rule with a power of -1.

1. Rewrite the equation:  
   \[
   y = (3x^2 - 5)^{-1}
   \]
2. Now, differentiate using the chain rule. The derivative of \( (3x^2 - 5)^{-1} \) is:  
   \[
   y' = -1 \cdot (3x^2 - 5)^{-2} \cdot (6x)
   \]
   Simplify:  
   \[
   y' = \frac{-6x}{(3x^2 - 5)^2}
   \]

### Problem 16:  
\[ y = x^{\frac{1}{3}} \]

We will differentiate this with respect to \( x \).

1. Apply the power rule:
   \[
   y' = \frac{1}{3}x^{\frac{1}{3} - 1} = \frac{1}{3}x^{-\frac{2}{3}}
   \]
   This is the derivative of \( y \).

### Problem 17:  
\[ y = \sqrt[4]{x(\sqrt{x} + 3)} \]

This expression is a bit more complex, and we will use both the product rule and chain rule.

1. First, rewrite the expression:
   \[
   y = (x(\sqrt{x} + 3))^{\frac{1}{4}}
   \]
2. Use the chain rule to differentiate the outer function:
   \[
   y' = \frac{1}{4}(x(\sqrt{x} + 3))^{-\frac{3}{4}} \cdot \frac{d}{dx}[x(\sqrt{x} + 3)]
   \]
3. Differentiate \( x(\sqrt{x} + 3) \) using the product rule:
   \[
   \frac{d}{dx}[x(\sqrt{x} + 3)] = \sqrt{x} + 3 + x \cdot \frac{1}{2\sqrt{x}} = \sqrt{x} + 3 + \frac{x}{2\sqrt{x}}
   \]
4. Combine the results for the full derivative.

### Problem 18:  
\[ y = \frac{x^3}{3x - 2} \]

We will use the quotient rule to differentiate this.

1. The quotient rule is \( \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \).

2. Let \( f(x) = x^3 \) and \( g(x) = 3x - 2 \).

3. Differentiate both:
   \[
   f'(x) = 3x^2, \quad g'(x) = 3
   \]
4. Apply the quotient rule:
   \[
   y' = \frac{(3x^2)(3x - 2) - (x^3)(3)}{(3x - 2)^2}
   \]
5. Simplify the result.

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Would you like detailed steps for any of these solutions or explanations for the derivatives? Here are some related questions for further practice:

1. How do you apply the chain rule to functions with nested radicals?
2. How does the product rule work when differentiating the product of two functions?
3. What is the difference between using the quotient rule and product rule in derivatives?
4. How do you simplify complex derivatives after applying differentiation rules?
5. What are some common mistakes when using the chain rule on radical expressions?

**Tip**: Always simplify the function as much as possible before differentiating—it often makes the process more straightforward!

Find the derivatives of the following functions: y = 1 / (3x^2 - 5), y = x^(1/3), y = 4th root of x(sqrt(x) + 3), y = x^3 / (3x - 2).

Learn how to differentiate complex functions using quotient, product, and chain rules. Examples include y = 1 / (3x^2 - 5), y = x^(1/3), y = 4th root of x(sqrt(x) + 3), and y = x^3 / (3x - 2). Step-by-step solutions provided.

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