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The problem asks to find the derivative of the function:

\[
y = \sqrt{x}(3x - 4)
\]

### Step-by-step solution:

We need to apply both the product rule and chain rule to differentiate this expression.

1. **Rewrite the function** to make it easier to differentiate:
   \[
   y = x^{1/2}(3x - 4)
   \]

2. **Apply the product rule** for differentiation: 
   \[
   \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)
   \]
   Here, \( f(x) = x^{1/2} \) and \( g(x) = 3x - 4 \).

3. **Differentiate each part**:
   - \( f(x) = x^{1/2} \) so \( f'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \)
   - \( g(x) = 3x - 4 \), so \( g'(x) = 3 \)

4. **Apply the product rule**:
   \[
   \frac{dy}{dx} = f'(x)g(x) + f(x)g'(x)
   \]
   \[
   \frac{dy}{dx} = \frac{1}{2\sqrt{x}}(3x - 4) + \sqrt{x}(3)
   \]

5. **Simplify the expression**:
   \[
   \frac{dy}{dx} = \frac{3x - 4}{2\sqrt{x}} + 3\sqrt{x}
   \]

6. **Combine terms if needed**:
   You can leave the answer in this form or express both terms with a common denominator if preferred.

### Final result:
\[
\frac{dy}{dx} = \frac{3x - 4}{2\sqrt{x}} + 3\sqrt{x}
\]

Would you like to see a more detailed explanation of any step? 

Here are some related questions you can explore:
1. How do you find the derivative using the chain rule?
2. What is the product rule in differentiation?
3. How do you simplify terms involving square roots?
4. What is the importance of the derivative in real-world applications?
5. How can the power rule be applied to fractional exponents?

**Tip:** When using the product rule, ensure that you clearly identify each function and its derivative before applying the rule to avoid mistakes.

Знайдіть похідну функції y = √x(3x - 4)

Learn how to differentiate the function y = √x(3x - 4) using the product rule and chain rule. Step-by-step solution includes applying basic calculus principles to find the derivative of a product of two functions. Suitable for high school or college students studying calculus.

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