Applications of Derivatives
Derivatives of product of polynomials
nta_pyq_2023_jan
Grade 12
Question:
Let $y(x) = (1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})$. Then $y' - y''$ at $x = -1$ is equal to
(1) 976
(2) 464
(3) 496
(4) 944
Step-by-Step Solution
Key Concept: Multiply: $y = \frac{1-x^{32}}{1-x}$ for $x \ne 1$. Differentiate implicitly: $y - xy = 1 - x^{32}$, giving $y' - xy' - y = -32x^{31}$ and $y'' - xy'' - y' - y' = -(32)(31)x^{30}$. Evaluate at $x = -1$.
We have $y(1-x) = 1 - x^{32}$. Differentiating: $y' - xy' - y = -32x^{31}$. Differentiating again: $y'' - xy'' - 2y' = -(32)(31)x^{30}$. At $x = -1$: $y(-1) = 0$, from first eq: $y'(-1) + y'(-1) - 0 = 32 \Rightarrow y'(-1) = 16$. From second eq: $y''(-1) + y''(-1) - 2(16) = -(32)(31) \Rightarrow 2y''(-1) = -992 + 32 = -960 \Rightarrow y''(-1) = -480$. Thus $y' - y'' = 16 - (-480) = 496$. Answer: (3).
Correct Answer: 3