The problem asks to determine which of the given expressions represents the derivative of a differentiable function $f$ at a point $b$.

Given the three expressions:

1. $\lim_{x \to b} \frac{f(b) - f(x)}{b - x}$
2. $\lim_{h \to 0} \frac{f(b - h) - f(b)}{h}$
3. $\lim_{h \to 0} \frac{f(b + h) - f(b - h)}{2h}$

Let's analyze each:

1. Expression I:
   This is the limit definition of the derivative, but with $b - x$ instead of $x - b$. Since the sign is reversed, this represents the derivative at $b$ but using the limit from the left. This expression is valid and gives $f'(b)$.

2. Expression II:
   This is just a re-arranged version of the limit definition of the derivative: $f'(b) = \lim_{h \to 0} \frac{f(b) - f(b - h)}{h}$ This also represents the derivative at $b$ but considers the limit as $h$ approaches 0 from the right.

3. Expression III:
   This expression is the symmetric difference quotient: $\frac{f(b + h) - f(b - h)}{2h}$ This also gives the derivative at $b$, and it is a commonly used expression because it averages the rate of change from both sides.

Conclusion:

All three expressions can be used to represent $f'(b)$, depending on the context or interpretation of the derivative.

Answer: All of them, I, II, and III, are correct representations of $f'(b)$.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:

1. How do the right-hand and left-hand derivatives relate to the overall derivative?
2. Why is the symmetric difference quotient useful in numerical differentiation?
3. What is the significance of the limit process in defining derivatives?
4. How can you prove that a function is differentiable at a point?
5. What happens if the limit does not exist for any of the expressions provided?

Tip: When dealing with derivatives, it is important to consider the direction from which the limit is taken, as this can affect the interpretation of the result.