1. **State the problem:** We are given the equation $$2kv(v+w) - (1 + k(v+w)^2) = 0$$ and want to understand or simplify it after differentiation. 2. **Rewrite the equation:** The equation is $$2kv(v+w) - (1 + k(v+w)^2) = 0$$. 3. **Expand terms:** Expand the terms inside the parentheses: $$2kv(v+w) = 2kv^2 + 2kvw$$ $$k(v+w)^2 = k(v^2 + 2vw + w^2)$$ 4. **Rewrite the equation with expansions:** $$2kv^2 + 2kvw - 1 - k(v^2 + 2vw + w^2) = 0$$ 5. **Distribute the minus sign:** $$2kv^2 + 2kvw - 1 - kv^2 - 2kvw - kw^2 = 0$$ 6. **Combine like terms:** $$ (2kv^2 - kv^2) + (2kvw - 2kvw) - kw^2 - 1 = 0$$ $$ kv^2 + 0 - kw^2 - 1 = 0$$ 7. **Simplify:** $$ kv^2 - kw^2 - 1 = 0$$ 8. **Rearrange:** $$ kv^2 - kw^2 = 1$$ 9. **Factor out k:** $$ k(v^2 - w^2) = 1$$ 10. **Solve for k:** $$ k = \frac{1}{v^2 - w^2}$$ **Final answer:** $$k = \frac{1}{v^2 - w^2}$$