Method of Differentiation
Differentiation of Products
IIT-JEE 2005
Grade 12
Question:
If $y = (1 + x)(1 + x^2)(1 + x^4)\cdots(1 + x^{2^n})$, then the value of $y'(0)$ is:
(1) $0$
(2) $1$
(3) $n$
(4) $2^n$
Step-by-Step Solution
Key Concept: Multiply the right side by $(1 - x)$ to telescope the product into $\frac{1 - x^{2^{n+1}}}{1 - x}$, which expands as a simple geometric series near $x = 0$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)