1. **State the problem:** Find the derivative of the function $$f(x) = \frac{3x^5 - 2x^2 + 1}{x^4 + x}$$ using the quotient rule. 2. **Recall the quotient rule formula:** For a function $$f(x) = \frac{g(x)}{h(x)}$$, the derivative is $$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$$ where $$g(x) = 3x^5 - 2x^2 + 1$$ and $$h(x) = x^4 + x$$. 3. **Find the derivatives of numerator and denominator:** - $$g'(x) = \frac{d}{dx}(3x^5 - 2x^2 + 1) = 15x^4 - 4x$$ - $$h'(x) = \frac{d}{dx}(x^4 + x) = 4x^3 + 1$$ 4. **Apply the quotient rule:** $$f'(x) = \frac{(15x^4 - 4x)(x^4 + x) - (3x^5 - 2x^2 + 1)(4x^3 + 1)}{(x^4 + x)^2}$$ 5. **Expand the terms in the numerator:** - First term: $$ (15x^4 - 4x)(x^4 + x) = 15x^8 + 15x^5 - 4x^5 - 4x^2 = 15x^8 + (15x^5 - 4x^5) - 4x^2 = 15x^8 + 11x^5 - 4x^2 $$ - Second term: $$ (3x^5 - 2x^2 + 1)(4x^3 + 1) = 3x^5 \cdot 4x^3 + 3x^5 \cdot 1 - 2x^2 \cdot 4x^3 - 2x^2 \cdot 1 + 1 \cdot 4x^3 + 1 \cdot 1 $$ $$= 12x^8 + 3x^5 - 8x^5 - 2x^2 + 4x^3 + 1 = 12x^8 + (3x^5 - 8x^5) - 2x^2 + 4x^3 + 1 = 12x^8 - 5x^5 - 2x^2 + 4x^3 + 1 $$ 6. **Subtract the second term from the first in the numerator:** $$ 15x^8 + 11x^5 - 4x^2 - (12x^8 - 5x^5 - 2x^2 + 4x^3 + 1) $$ $$= 15x^8 + 11x^5 - 4x^2 - 12x^8 + 5x^5 + 2x^2 - 4x^3 - 1 $$ $$= (15x^8 - 12x^8) + (11x^5 + 5x^5) + (-4x^2 + 2x^2) - 4x^3 - 1 $$ $$= 3x^8 + 16x^5 - 2x^2 - 4x^3 - 1 $$ 7. **Write the final derivative:** $$f'(x) = \frac{3x^8 + 16x^5 - 2x^2 - 4x^3 - 1}{(x^4 + x)^2}$$ This is the derivative of the given function using the quotient rule.