1. **Problem:** Given $y = 3x^4 - 6x^2 + x - 4$, find (i) $y'(2)$ and (ii) $y'''(2)$. 2. **Step 1: Differentiate $y$ to find $y'$** Use the power rule: $\frac{d}{dx} x^n = n x^{n-1}$. $$y' = \frac{d}{dx}(3x^4) - \frac{d}{dx}(6x^2) + \frac{d}{dx}(x) - \frac{d}{dx}(4) = 12x^3 - 12x + 1$$ 3. **Step 2: Evaluate $y'(2)$** $$y'(2) = 12(2)^3 - 12(2) + 1 = 12 \times 8 - 24 + 1 = 96 - 24 + 1 = 73$$ 4. **Step 3: Find $y''$ by differentiating $y'$** $$y'' = \frac{d}{dx}(12x^3 - 12x + 1) = 36x^2 - 12$$ 5. **Step 4: Find $y'''$ by differentiating $y''$** $$y''' = \frac{d}{dx}(36x^2 - 12) = 72x$$ 6. **Step 5: Evaluate $y'''(2)$** $$y'''(2) = 72 \times 2 = 144$$ 7. **Summary:** - $y'(2) = 73$ - $y'''(2) = 144$ Note: The user stated $y'(2) = 83$ and $y'''(2) = 132$ but the correct calculations yield $73$ and $144$ respectively.