https://file.aiquickdraw.com/mathbot/file/1729694753727jefkdxc6.jpg

Let's solve both derivatives from the image.

### 1. \( h(x) = \sqrt{x} + \frac{3}{x} \)

To find the derivative \( h'(x) \), we differentiate each term individually:

- Derivative of \( \sqrt{x} \) is:
\[
\frac{d}{dx} (\sqrt{x}) = \frac{1}{2\sqrt{x}}
\]

- Derivative of \( \frac{3}{x} \) is:
\[
\frac{d}{dx} \left( \frac{3}{x} \right) = -\frac{3}{x^2}
\]

Thus, the derivative \( h'(x) \) is:
\[
h'(x) = \frac{1}{2\sqrt{x}} - \frac{3}{x^2}
\]

---

### 2. \( f(x) = \frac{x + 2}{x^3 - 1} \)

For this, we'll use the quotient rule for derivatives, which states:

\[
\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}
\]

Here, \( u = x + 2 \) and \( v = x^3 - 1 \).

- Derivative of \( u = x + 2 \) is:
\[
u' = 1
\]

- Derivative of \( v = x^3 - 1 \) is:
\[
v' = 3x^2
\]

Now, applying the quotient rule:
\[
f'(x) = \frac{(1)(x^3 - 1) - (x + 2)(3x^2)}{(x^3 - 1)^2}
\]

Simplifying the numerator:
\[
f'(x) = \frac{x^3 - 1 - 3x^2(x + 2)}{(x^3 - 1)^2}
= \frac{x^3 - 1 - 3x^3 - 6x^2}{(x^3 - 1)^2}
= \frac{-2x^3 - 6x^2 - 1}{(x^3 - 1)^2}
\]

Thus, the derivative \( f'(x) \) is:
\[
f'(x) = \frac{-2x^3 - 6x^2 - 1}{(x^3 - 1)^2}
\]

---

Let me know if you want further details, and here are some related questions for further exploration:

1. How is the chain rule applied when differentiating composite functions?
2. Can you apply the product rule and quotient rule together in a single problem?
3. What is the general rule for differentiating powers of \( x \)?
4. How would you differentiate more complex functions like trigonometric or exponential functions?
5. How do derivatives help in finding the slope of a curve at a specific point?

**Tip:** Always simplify your function as much as possible before applying the derivative rules; this can help avoid unnecessary complications.

Find the derivatives of h(x) = √x + 3/x and f(x) = (x + 2)/(x^3 - 1).

Learn how to calculate the derivatives of two functions: h(x) = √x + 3/x and f(x) = (x + 2)/(x^3 - 1) using basic differentiation rules such as the power rule and the quotient rule. This problem is suitable for high school calculus students.

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