1. The problem states that $y = vw$ and asks to identify which given expression for $y_n$ (where subscripts denote derivatives) is false. 2. We interpret $y = vw$ as a product of two functions $v$ and $w$. Using the Leibniz rule for the $n$th derivative of a product: $$ y_n = \sum_{k=0}^n \binom{n}{k} v^{(k)} w^{(n-k)} $$ where $v^{(k)}$ denotes the $k$th derivative of $v$ and similarly for $w^{(n-k)}$. 3. Verify each option using the formula for derivatives of products. - For $y_1$ (the first derivative): $$ y_1 = v w_1 + v_1 w $$ This matches option A, so A is true. - For $y_2$ (the second derivative): $$ y_2 = v w_2 + 2 v_1 w_1 + v_2 w $$ Option B says $y_2 = v w_2 + 2 w_1 v_1 + v_2 w$ which is the same expression (order of multiplication does not matter). So B is true. - For $y_3$ (the third derivative): $$ y_3 = v w_3 + 3 v_1 w_2 + 3 v_2 w_1 + v_3 w $$ Option C says $y_3 = v w_3 + 3 v_2 w_1 + v_3 w$ and misses the $3 v_1 w_2$ term, so C is false. - For $y_4$ (the fourth derivative): $$ y_4 = v w_4 + 4 v_1 w_3 + 6 v_2 w_2 + 4 v_3 w_1 + v_4 w $$ Option D matches this exactly, so D is true. Final answer: Option C is false. **Answer:** C is false because it omits the $3 v_1 w_2$ term in $y_3$.