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This problem requires calculating the derivative of each given function \( f(x) \) using the definition of the derivative:

\[
\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
\]

Let's go through each case.

### (a) \( f(x) = x^2 \)

1. Substitute \( f(x+h) = (x+h)^2 \) and \( f(x) = x^2 \).
   
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{(x+h)^2 - x^2}{h}
   \]

2. Expand \( (x+h)^2 = x^2 + 2xh + h^2 \):

   \[
   = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = \frac{h(2x + h)}{h} = 2x + h.
   \]

3. Take the limit as \( h \to 0 \):

   \[
   \lim_{h \to 0} (2x + h) = 2x.
   \]

So, \( f'(x) = 2x \).

---

### (b) \( f(x) = 5 \)

1. Since \( f(x+h) = 5 \) and \( f(x) = 5 \), the expression becomes:

   \[
   \frac{f(x+h) - f(x)}{h} = \frac{5 - 5}{h} = \frac{0}{h} = 0.
   \]

2. The limit as \( h \to 0 \) is simply 0.

So, \( f'(x) = 0 \).

---

### (c) \( f(x) = 2x^2 + x \)

1. Substitute \( f(x+h) = 2(x+h)^2 + (x+h) \) and \( f(x) = 2x^2 + x \):

   \[
   \frac{f(x+h) - f(x)}{h} = \frac{2(x+h)^2 + (x+h) - (2x^2 + x)}{h}
   \]

2. Expand \( 2(x+h)^2 = 2(x^2 + 2xh + h^2) = 2x^2 + 4xh + 2h^2 \) and \( (x+h) = x + h \):

   \[
   = \frac{(2x^2 + 4xh + 2h^2 + x + h) - (2x^2 + x)}{h}
   \]

3. Simplify terms:

   \[
   = \frac{4xh + 2h^2 + h}{h} = \frac{h(4x + 2h + 1)}{h} = 4x + 2h + 1.
   \]

4. Take the limit as \( h \to 0 \):

   \[
   \lim_{h \to 0} (4x + 2h + 1) = 4x + 1.
   \]

So, \( f'(x) = 4x + 1 \).

---

### Summary of Results

- (a) \( f(x) = x^2 \Rightarrow f'(x) = 2x \)
- (b) \( f(x) = 5 \Rightarrow f'(x) = 0 \)
- (c) \( f(x) = 2x^2 + x \Rightarrow f'(x) = 4x + 1 \)

Would you like further details on any of these steps, or do you have any questions?

---

### Related Questions

1. What is the derivative of \( f(x) = 3x^2 + 2x + 1 \) using the same limit definition?
2. How do derivatives represent the slope of a function at a given point?
3. What is the physical interpretation of the derivative in terms of velocity, if \( f(x) \) represents distance?
4. How does the definition of the derivative apply to trigonometric functions, such as \( f(x) = \sin(x) \)?
5. Can you explain how the derivative of a constant function is always zero?

### Tip
When applying the limit definition of the derivative, simplifying algebraic expressions is key before taking the limit as \( h \to 0 \).

Calculate lim (h->0) [f(x+h) - f(x)] / h for each given function: (a) f(x) = x^2, (b) f(x) = 5, (c) f(x) = 2x^2 + x.

Learn how to calculate the derivative of functions using the limit definition of derivatives. Step-by-step solutions are provided for f(x) = x^2, f(x) = 5, and f(x) = 2x^2 + x, covering key calculus concepts of limits and differentiation.

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