1. Problem: Find the derivative of $f(x) = \frac{3}{4}$ using the constant rule. The constant rule states that the derivative of a constant is zero. $$f'(x) = 0$$ 2. Problem: Find the derivative of $f(x) = x^9$ using the power rule. The power rule states that $\frac{d}{dx} x^n = nx^{n-1}$. $$f'(x) = 9x^{8}$$ 3. Problem: Find the derivative of $f(x) = 8x^7$ using the constant multiple and power rules. Using the constant multiple rule, $\frac{d}{dx} [cf(x)] = c \frac{d}{dx} f(x)$. $$f'(x) = 8 \cdot 7x^{6} = 56x^{6}$$ 4. Problem: Find the derivative of $f(x) = \frac{4}{3x^6}$. Rewrite as $f(x) = \frac{4}{3} x^{-6}$. Using constant multiple and power rules: $$f'(x) = \frac{4}{3} \cdot (-6) x^{-7} = -\frac{24}{3} x^{-7} = -8x^{-7}$$ 5. Problem: Find the derivative of $f(x) = \frac{2}{\sqrt{x^3}}$. Rewrite $\sqrt{x^3} = x^{3/2}$, so $f(x) = 2x^{-3/2}$. Using constant multiple and power rules: $$f'(x) = 2 \cdot \left(-\frac{3}{2}\right) x^{-5/2} = -3x^{-5/2}$$ 6. Problem: Find the derivative of $f(x) = 2x^4 - 3x^3 + 5x^2 - 5x + 7$ using sum and difference rules. Derivative of each term: $$\frac{d}{dx} 2x^4 = 8x^3$$ $$\frac{d}{dx} (-3x^3) = -9x^2$$ $$\frac{d}{dx} 5x^2 = 10x$$ $$\frac{d}{dx} (-5x) = -5$$ $$\frac{d}{dx} 7 = 0$$ Sum: $$f'(x) = 8x^3 - 9x^2 + 10x - 5$$ 7. Problem: Find the derivative of $f(x) = \frac{12}{x^4} - \frac{x^3}{2} + x^2$. Rewrite terms: $$12x^{-4} - \frac{1}{2}x^3 + x^2$$ Derivatives: $$-48x^{-5} - \frac{3}{2}x^2 + 2x$$ 8. Problem: Find the derivative of $f(x) = (12x^4 + x^3)(x^2 + 10x)$ using the product rule. Product rule: $\frac{d}{dx}[u v] = u'v + uv'$. Let $u = 12x^4 + x^3$, $v = x^2 + 10x$. $$u' = 48x^3 + 3x^2$$ $$v' = 2x + 10$$ Apply product rule: $$f'(x) = (48x^3 + 3x^2)(x^2 + 10x) + (12x^4 + x^3)(2x + 10)$$ 9. Problem: Find the derivative of $f(x) = \frac{2x + 9}{3x + 4}$ using the quotient rule. Quotient rule: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}$. Let $u = 2x + 9$, $v = 3x + 4$. $$u' = 2$$ $$v' = 3$$ Apply quotient rule: $$f'(x) = \frac{2(3x + 4) - (2x + 9)(3)}{(3x + 4)^2}$$ Simplify numerator: $$= \frac{6x + 8 - (6x + 27)}{(3x + 4)^2} = \frac{6x + 8 - 6x - 27}{(3x + 4)^2} = \frac{-19}{(3x + 4)^2}$$ 10. Problem: Find the derivative of $f(x) = \sqrt{3} - x - x^2$. Derivative of constants and powers: $$\frac{d}{dx} \sqrt{3} = 0$$ $$\frac{d}{dx} (-x) = -1$$ $$\frac{d}{dx} (-x^2) = -2x$$ Sum: $$f'(x) = -1 - 2x$$