Find the derivatives of the following at any point of their domains :
`f(x)=(x-a)(x-b)`

Answer

To find the derivative of the function \( f(x) = (x - a)(x - b) \), we will use the product rule of differentiation. The product rule states that if you have a function that is the product of two functions, say \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ (uv)' = u'v + uv' \]

### Step-by-step solution:

1. **Identify the functions**:
Let \( u = (x - a) \) and \( v = (x - b) \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) is:
\[ u' = \frac{d}{dx}(x - a) = 1 \]
- The derivative of \( v \) is:
\[ v' = \frac{d}{dx}(x - b) = 1 \]

3. **Apply the product rule**:
Now, using the product rule:
\[ f'(x) = u'v + uv' \]
Substituting the values we found:
\[ f'(x) = (1)(x - b) + (x - a)(1) \]

4. **Simplify the expression**:
\[ f'(x) = (x - b) + (x - a) \]
\[ f'(x) = x - b + x - a \]
\[ f'(x) = 2x - (a + b) \]

### Final Answer:
Thus, the derivative of the function \( f(x) = (x - a)(x - b) \) is:
\[ f'(x) = 2x - (a + b) \]