1. **Problem statement:** Differentiate the function $f(x) = \frac{u(x)}{v(x)}$ using the quotient rule. 2. **Formula:** The quotient rule states: $$f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$$ where $u(x)$ and $v(x)$ are differentiable functions. 3. **Explanation:** To find the derivative of a quotient, you take the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. 4. **Intermediate work:** - Compute $u'(x)$, the derivative of the numerator. - Compute $v'(x)$, the derivative of the denominator. - Substitute into the formula. - Simplify the expression if possible. 5. **Final answer:** $$f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$$ This is the derivative of the quotient $\frac{u(x)}{v(x)}$ using the quotient rule.