1. The problem is to understand and apply the quotient rule in calculus to differentiate a function that is the ratio of two functions. 2. The quotient rule formula is: $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$$ 3. Important rules: - $f(x)$ and $g(x)$ must be differentiable. - $g(x) \neq 0$ to avoid division by zero. 4. To apply the quotient rule: - Differentiate the numerator function $f(x)$ to get $f'(x)$. - Differentiate the denominator function $g(x)$ to get $g'(x)$. - Substitute into the formula. - Simplify the expression if possible. 5. Example: Differentiate $h(x) = \frac{x^2 + 1}{x - 3}$. - $f(x) = x^2 + 1$, so $f'(x) = 2x$. - $g(x) = x - 3$, so $g'(x) = 1$. - Apply quotient rule: $$h'(x) = \frac{2x(x - 3) - (x^2 + 1)(1)}{(x - 3)^2}$$ - Simplify numerator: $$2x^2 - 6x - x^2 - 1 = x^2 - 6x - 1$$ - Final derivative: $$h'(x) = \frac{x^2 - 6x - 1}{(x - 3)^2}$$ This is the derivative of the quotient using the quotient rule.