If `f(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5)`, then the value of f' (5) is equal to

Options

A. 0
B. 120
C. 24
D. 5

Answer

To find the value of \( f'(5) \) for the function \( f(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5) \), we will use the product rule of differentiation.

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5) \]

2. **Differentiate using the product rule**:
The product rule states that if you have a product of functions, the derivative is given by:
\[ (u \cdot v)' = u'v + uv' \]
In our case, we have five factors. We can differentiate each factor one at a time while keeping the others constant.

3. **Differentiate each term**:
- Differentiate the first term \( (x - 1) \):
\[ f'(x) = 1 \cdot (x - 2)(x - 3)(x - 4)(x - 5) \]
- Differentiate the second term \( (x - 2) \):
\[ f'(x) = (x - 1) \cdot 1 \cdot (x - 3)(x - 4)(x - 5) \]
- Differentiate the third term \( (x - 3) \):
\[ f'(x) = (x - 1)(x - 2) \cdot 1 \cdot (x - 4)(x - 5) \]
- Differentiate the fourth term \( (x - 4) \):
\[ f'(x) = (x - 1)(x - 2)(x - 3) \cdot 1 \cdot (x - 5) \]
- Differentiate the fifth term \( (x - 5) \):
\[ f'(x) = (x - 1)(x - 2)(x - 3)(x - 4) \cdot 1 \]

4. **Combine all derivatives**:
Thus, we can express \( f'(x) \) as:
\[ f'(x) = (x - 2)(x - 3)(x - 4)(x - 5) + (x - 1)(x - 3)(x - 4)(x - 5) + (x - 1)(x - 2)(x - 4)(x - 5) + (x - 1)(x - 2)(x - 3)(x - 5) + (x - 1)(x - 2)(x - 3)(x - 4) \]

5. **Evaluate at \( x = 5 \)**:
Now, we need to find \( f'(5) \):
- When substituting \( x = 5 \) into each term:
- The first term becomes \( (5 - 2)(5 - 3)(5 - 4)(5 - 5) = 3 \cdot 2 \cdot 1 \cdot 0 = 0 \)
- The second term becomes \( (5 - 1)(5 - 3)(5 - 4)(5 - 5) = 4 \cdot 2 \cdot 1 \cdot 0 = 0 \)
- The third term becomes \( (5 - 1)(5 - 2)(5 - 4)(5 - 5) = 4 \cdot 3 \cdot 1 \cdot 0 = 0 \)
- The fourth term becomes \( (5 - 1)(5 - 2)(5 - 3)(5 - 5) = 4 \cdot 3 \cdot 2 \cdot 0 = 0 \)
- The fifth term becomes \( (5 - 1)(5 - 2)(5 - 3)(5 - 4) = 4 \cdot 3 \cdot 2 \cdot 1 = 24 \)

6. **Final Result**:
Therefore, adding these results together:
\[ f'(5) = 0 + 0 + 0 + 0 + 24 = 24 \]

Thus, the value of \( f'(5) \) is \( \boxed{24} \).