AP EAMCET · Maths · Differentiation
If \(f(x)=\frac{1}{1+\frac{1}{x}}\) and \(g(x)=\frac{1}{1+\frac{1}{f(x)}}\), then \(g^{\prime}(2)\) is equal to
- A \(\frac{1}{5}\)
- B \(\frac{1}{25}\)
- C \(5\)
- D \(\frac{1}{16}\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{25}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \quad f(x)=\frac{1}{1+\frac{1}{x}}=\frac{x}{1+x} \\ & \text { and } \quad g(x)=\frac{1}{1+\frac{1}{f(x)}}=\frac{1}{1+\frac{1+x}{x}}=\frac{x}{2 x+1} \\ & \therefore \quad g^{\prime}(x)=\frac{(2 x+1) \cdot 1-x(2)}{(2 x+1)^2}=\frac{1}{(2 x+1)^2} \\ & \text { Now,…
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