Let's go through the derivatives of the given functions step by step.

23. $f(x) = \frac{1}{x + 2}$

This is a quotient, and we can use the quotient rule or rewrite the function as $(x + 2)^{-1}$.

Using the chain rule: $f'(x) = -1 \cdot (x + 2)^{-2} \cdot \frac{d}{dx}(x + 2) = -\frac{1}{(x + 2)^2}$

24. $f(x) = x^2 - 3x + 4$

This is a polynomial function. We apply the power rule: $f'(x) = 2x - 3$

25. $g(x) = \frac{x}{x - 1}$

We apply the quotient rule here: $\left( \frac{u}{v} \right)' = \frac{v u' - u v'}{v^2}$ where $u = x$ and $v = x - 1$. Now:

• $u' = 1$
• $v' = 1$

Thus: $g'(x) = \frac{(x - 1)(1) - (x)(1)}{(x - 1)^2} = \frac{x - 1 - x}{(x - 1)^2} = \frac{-1}{(x - 1)^2}$

26. $g(x) = 1 + \sqrt{x}$

Rewrite $\sqrt{x}$ as $x^{1/2}$ and apply the power rule: $g'(x) = 0 + \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

Do you need further explanations on any of these solutions?

Related questions:

1. How does the product rule work, and how is it different from the quotient rule?
2. What is the chain rule, and when should you apply it?
3. How do you find higher-order derivatives of a function?
4. What happens to the derivative when the function contains logarithmic terms?
5. How do you apply implicit differentiation to solve equations involving multiple variables?

Tip: Always simplify functions before differentiating them. It can often save you time and reduce errors.