1. **State the problem:** We want to simplify the function $$y = \frac{x^2 - 49}{x^2 + 5x - 14}$$ to differentiate it, considering the domain restrictions where the denominator is not zero. 2. **Factor numerator and denominator:** - Numerator: $$x^2 - 49 = (x - 7)(x + 7)$$ (difference of squares). - Denominator: $$x^2 + 5x - 14 = (x + 7)(x - 2)$$ (factored quadratic). 3. **Simplify the expression:** $$y = \frac{(x - 7)(x + 7)}{(x + 7)(x - 2)}$$ Cancel the common factor $(x + 7)$, but note that $x \neq -7$ to avoid division by zero. So, $$y = \frac{x - 7}{x - 2}, \quad x \neq -7, x \neq 2$$ 4. **Domain restrictions:** - The original denominator $x^2 + 5x - 14$ must not be zero, so $x \neq -7$ and $x \neq 2$. - After simplification, these restrictions remain. 5. **Ready to differentiate:** The simplified function is $$y = \frac{x - 7}{x - 2}$$ with domain $x \neq -7, 2$. This form is easier to differentiate using the quotient rule or rewriting as a product. **Final simplified function:** $$y = \frac{x - 7}{x - 2}$$ with domain $x \neq -7, 2$.