Derivatives R Cubed 842A62

1. **State the problem:** We need to find the second derivative $y''$ and the third derivative $y'''$ of the function $$y = \frac{2}{3} \pi r^3$$ with respect to $r$. 2. **Recall the formula for derivatives:** The derivative of $r^n$ with respect to $r$ is $$\frac{d}{dr} r^n = n r^{n-1}$$. 3. **First derivative:** $$y' = \frac{d}{dr} \left( \frac{2}{3} \pi r^3 \right) = \frac{2}{3} \pi \cdot 3 r^{3-1} = 2 \pi r^2$$ 4. **Second derivative:** $$y'' = \frac{d}{dr} y' = \frac{d}{dr} (2 \pi r^2) = 2 \pi \cdot 2 r^{2-1} = 4 \pi r$$ 5. **Third derivative:** $$y''' = \frac{d}{dr} y'' = \frac{d}{dr} (4 \pi r) = 4 \pi \cdot 1 r^{1-1} = 4 \pi$$ **Final answers:** $$y'' = 4 \pi r$$ $$y''' = 4 \pi$$