Question

If \[ f(x)=(1+x^3)(1+x^6)(1+x^{12})(1+x^{24}) \] then \(f'(-1)=\)

• A. \(24\)
• B. \(12\)
• C. \(48\)
• D. \(60\)

Hint:

Whenever powers double repeatedly (\(x^3,x^6,x^{12}\)), search immediately for telescoping product identities.

Answer 1

The Correct Option is C

Concept: Use logarithmic differentiation or algebraic identity. Notice pattern: \[ (1-x)(1+x)=1-x^2 \] Repeated telescoping often simplifies expressions.

Step 1: Observe product pattern.
Using identity \[ (1-x)(1+x)(1+x^2)(1+x^4)\dots = 1-x^{2^n} \] Transforming expression: \[ f(x)=\frac{1-x^{48}}{1-x^3} \]

Step 2: Differentiate.
Using quotient rule. \[ f'(x) = \frac{-48x^{47}(1-x^3)+(1-x^{48})(3x^2)} {(1-x^3)^2} \]

Step 3: Substitute \(x=-1\).
After simplification: \[ f'(-1)=48 \] Thus \[ \boxed{48} \]