Differentiation Question 2
Question 2 - 25 January - Shift 1
Let
$y(x)=(1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})$
Then $y^{\prime}-y^{\prime \prime}$ at $x=-1$ is equal to
(1) 976
(2) 464
(3) 496
(4) 944
Show Answer
Answer: (3)
Solution:
Formula: Sum of G.P, Implicit differentiation
$y(x)=(1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})$
$y(x)=(1+x)[(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})]$
$y(x)=(1+x)[(1+x^{2}+x^{4}+x^{6}+x^{8}+x^{10}+x^{12}+…..+x^{32})]$
$ \begin{aligned} & y=\frac{1-x^{32}}{1-x^{2}} \\ & y=\frac{1-x^{32}}{1-x} \\ & \Rightarrow y-x y=1-x^{32} \\ & y^{\prime}-x y^{\prime}-y=-32 x^{31} \\ & y^{\prime \prime}-x y^{\prime \prime}-y^{\prime}-y^{\prime}=-(32)(31) x^{30} \\ & \text{ at } x=-1 \Rightarrow y^{\prime}-y^{\prime \prime}=496 \end{aligned} $
