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Tentukan turunan pertama dari fungsi berikut 

Let's solve the first derivatives for the functions provided in the image:

### 8. \( y = \frac{x^2}{2} + \sqrt{3x} - \frac{2}{x} + 2x \)

- First term: \( \frac{x^2}{2} \). Using the power rule \( \frac{d}{dx} x^n = n x^{n-1} \), the derivative is:
  \[
  \frac{d}{dx} \left( \frac{x^2}{2} \right) = \frac{2x}{2} = x
  \]
  
- Second term: \( \sqrt{3x} = (3x)^{1/2} \). Use the chain rule:
  \[
  \frac{d}{dx} \left( (3x)^{1/2} \right) = \frac{1}{2} (3x)^{-1/2} \cdot 3 = \frac{3}{2} \cdot \frac{1}{\sqrt{3x}} = \frac{3}{2\sqrt{3x}}
  \]
  
- Third term: \( -\frac{2}{x} \). The derivative of \( x^{-1} \) is \( -x^{-2} \):
  \[
  \frac{d}{dx} \left( -\frac{2}{x} \right) = 2x^{-2} = \frac{2}{x^2}
  \]

- Fourth term: \( 2x \). The derivative of \( 2x \) is just \( 2 \).

Thus, the derivative of the function is:
\[
y' = x + \frac{3}{2\sqrt{3x}} + \frac{2}{x^2} + 2
\]

---

### 9. \( y = (3x - 2)(x^2 + 3) \)

Use the product rule: \( \frac{d}{dx} (u \cdot v) = u'v + uv' \).

- Let \( u = 3x - 2 \) and \( v = x^2 + 3 \).
  - \( u' = 3 \)
  - \( v' = 2x \)

Now, apply the product rule:
\[
y' = (3)(x^2 + 3) + (3x - 2)(2x)
\]
\[
y' = 3(x^2 + 3) + (3x - 2)(2x)
\]
Expanding both terms:
\[
y' = 3x^2 + 9 + 6x^2 - 4x
\]
Combine like terms:
\[
y' = 9x^2 - 4x + 9
\]

---

### 10. \( y = \frac{x + 1}{\sqrt{x}} \)

This can be written as \( y = (x + 1)x^{-1/2} \), and we apply the product rule.

Let \( u = x + 1 \) and \( v = x^{-1/2} \).
- \( u' = 1 \)
- \( v' = -\frac{1}{2}x^{-3/2} \)

Now apply the product rule:
\[
y' = (x + 1) \left( -\frac{1}{2}x^{-3/2} \right) + x^{-1/2}
\]
\[
y' = -\frac{x+1}{2x^{3/2}} + \frac{1}{x^{1/2}}
\]

Thus, the derivative is:
\[
y' = \frac{1}{\sqrt{x}} - \frac{x+1}{2x^{3/2}}
\]

---

Would you like more details or clarifications on these steps?

Here are 5 related questions to explore:
1. How do you apply the product rule in a derivative?
2. How does the chain rule affect derivatives of composite functions?
3. How do you find the derivative of a function with square roots?
4. What is the importance of combining like terms in derivatives?
5. How do you handle negative exponents in differentiation?

**Tip:** When dealing with multiple terms, always treat each term independently and then combine the results carefully to avoid errors.

Tentukan turunan pertama dari fungsi berikut: y = x^2/2 + √3x - 2/x + 2x, y = (3x - 2)(x^2 + 3), y = (x + 1)/√x

Learn how to find the first derivative of functions like y = x^2/2 + √3x - 2/x + 2x, y = (3x - 2)(x^2 + 3), and y = (x + 1)/√x using the power, product, and chain rules. This step-by-step guide covers differentiation and algebraic techniques, ideal for students in Grades 10-12.

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