Question

For some constants \( a \) and \( b \), find the derivative of \( \frac{x - a}{x - b} \)

Hint:

Use the quotient rule for derivatives when you have a ratio of two functions.

Answer 1

Step 1: Define the function.
We are given the function: \[ f(x) = \frac{x - a}{x - b} \]
Step 2: Apply the quotient rule.
The quotient rule for differentiation states that for a function of the form \( \frac{u(x)}{v(x)} \), the derivative is: \[ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{v(x)^2} \] In our case:
- \( u(x) = x - a \), so \( u'(x) = 1 \)
- \( v(x) = x - b \), so \( v'(x) = 1 \)

Step 3: Compute the derivative.
Using the quotient rule: \[ f'(x) = \frac{(x - b) \cdot 1 - (x - a) \cdot 1}{(x - b)^2} \] Simplify the numerator: \[ f'(x) = \frac{x - b - x + a}{(x - b)^2} \] \[ f'(x) = \frac{a - b}{(x - b)^2} \]