1. **Problem:** Find the first and second derivatives of $f(x) = x^2 + x + 8$. Formula: For $f(x) = x^n$, $f'(x) = nx^{n-1}$ and $f''(x) = n(n-1)x^{n-2}$. Step 1: First derivative: $$f'(x) = 2x + 1 + 0 = 2x + 1$$ Step 2: Second derivative: $$f''(x) = 2 + 0 = 2$$ 2. **Problem:** Find the first and second derivatives of $f(x) = 5x^3 - 3x^5$. Step 1: First derivative: $$f'(x) = 15x^2 - 15x^4$$ Step 2: Second derivative: $$f''(x) = 30x - 60x^3$$ 3. **Problem:** Find the first and second derivatives of $f(x) = \frac{x^3 + 7}{x}$. Rewrite: $$f(x) = x^2 + \frac{7}{x} = x^2 + 7x^{-1}$$ Step 1: First derivative: $$f'(x) = 2x - 7x^{-2} = 2x - \frac{7}{x^2}$$ Step 2: Second derivative: $$f''(x) = 2 + 14x^{-3} = 2 + \frac{14}{x^3}$$ 4. **Problem:** Find the first and second derivatives of $f(x) = (x+1)(x-1)$. Step 1: Expand: $$f(x) = x^2 - 1$$ Step 2: First derivative: $$f'(x) = 2x$$ Step 3: Second derivative: $$f''(x) = 2$$ 5. **Problem:** Find the first and second derivatives of $f(t) = \frac{t^2 + 5t - 1}{t^2}$. Rewrite: $$f(t) = 1 + 5t t^{-2} - t^{-2} = 1 + 5t^{-1} - t^{-2}$$ Step 1: First derivative: $$f'(t) = 0 - 5t^{-2} + 2t^{-3} = -\frac{5}{t^2} + \frac{2}{t^3}$$ Step 2: Second derivative: $$f''(t) = 10t^{-3} - 6t^{-4} = \frac{10}{t^3} - \frac{6}{t^4}$$ 6. **Problem:** Find the first and second derivatives of $f(x) = \frac{1}{2}x^2 + 3x - 4$. Step 1: First derivative: $$f'(x) = x + 3$$ Step 2: Second derivative: $$f''(x) = 1$$ 7. **Problem:** Find the first and second derivatives of $f(x) = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4}$. Step 1: First derivative: $$f'(x) = x^2 + x + \frac{1}{4}$$ Step 2: Second derivative: $$f''(x) = 2x + 1$$