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 \( f(x) = \frac{x-a}{x-b} \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function \( f(x) = \frac{u}{v} \), then the derivative \( f'(x) \) is given by:

\[ f'(x) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]

where \( u = x - a \) and \( v = x - b \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x - a \)
- \( v = x - b \)

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
Now, we differentiate \( u \) and \( v \):
- \( \frac{du}{dx} = \frac{d}{dx}(x - a) = 1 \) (since the derivative of \( x \) is 1 and the derivative of a constant is 0)
- \( \frac{dv}{dx} = \frac{d}{dx}(x - b) = 1 \)

### Step 3: Apply the Quotient Rule
Using the quotient rule:
\[ f'(x) = \frac{(x - b) \cdot 1 - (x - a) \cdot 1}{(x - b)^2} \]

### Step 4: Simplify the expression
Now we simplify the numerator:
\[ f'(x) = \frac{(x - b) - (x - a)}{(x - b)^2} \]
\[ = \frac{x - b - x + a}{(x - b)^2} \]
\[ = \frac{a - b}{(x - b)^2} \]

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