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.