To find the derivative of the function $y = \frac{x^2 - 3x + 2}{x^7 - 2}$, we will apply the quotient rule. The quotient rule states that if you have a function in the form of $y = \frac{f(x)}{g(x)}$, the derivative is:

$\frac{dy}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

In this case, we define:

• $f(x) = x^2 - 3x + 2$
• $g(x) = x^7 - 2$

Now, let's calculate the derivatives of $f(x)$ and $g(x)$:

• $f'(x) = 2x - 3$
• $g'(x) = 7x^6$

Using the quotient rule:

$\frac{dy}{dx} = \frac{(2x - 3)(x^7 - 2) - (x^2 - 3x + 2)(7x^6)}{(x^7 - 2)^2}$

Now, simplify the numerator:

= 2x^8 - 4x - 3x^7 + 6$$ $$(x^2 - 3x + 2)(7x^6) = 7x^6(x^2 - 3x + 2) = 7x^8 - 21x^7 + 14x^6$$ So, the numerator becomes: $$(2x^8 - 4x - 3x^7 + 6) - (7x^8 - 21x^7 + 14x^6) = 2x^8 - 4x - 3x^7 + 6 - 7x^8 + 21x^7 - 14x^6 = -5x^8 + 18x^7 - 14x^6 - 4x + 6$$ Thus, the derivative is: $$\frac{dy}{dx} = \frac{-5x^8 + 18x^7 - 14x^6 - 4x + 6}{(x^7 - 2)^2}$$ ### Summary: $$\frac{dy}{dx} = \frac{-5x^8 + 18x^7 - 14x^6 - 4x + 6}{(x^7 - 2)^2}$$ Would you like to explore any specific details or further simplification? Here are 5 related questions you could explore: 1. How do you apply the quotient rule when both the numerator and denominator are polynomials? 2. Can the quotient rule be simplified when the numerator and denominator have common factors? 3. How does the derivative change when the function involves higher powers of $$x$$? 4. How do you find critical points of a function using its derivative? 5. What does the second derivative of this function tell us about its concavity? **Tip:** Remember, when using the quotient rule, always keep track of both terms carefully in the numerator and denominator to avoid sign errors!