AP EAMCET · Maths · Differentiation
If \(a>0\) and \(f(x)=\left(\frac{a+x}{1+x}\right)^{a+1+2 x}\), then \(f^{\prime}(0)=\)
- A \(a^{a+1}\)
- B \(a^{a+1}\left\{\frac{1-a^2}{a}+2 \log a\right\}\)
- C \(2 \log a\)
- D \(a^{a+1}\left\{\frac{(1+a)^2}{a-2 \log a}\right\}\)
Answer & Solution
Correct Answer
(B) \(a^{a+1}\left\{\frac{1-a^2}{a}+2 \log a\right\}\)
Step-by-step Solution
Detailed explanation
Given function, for \(a>0\) \[ \begin{aligned} & f(x)=\left(\frac{a+x}{1+x}\right)^{a+1+2 x} \Rightarrow f(0)=a^{a+1} \\ \Rightarrow \quad & \log f(x)=(a+1+2 x) \log \left(\frac{a+x}{1+x}\right) \end{aligned} \] On differentiating both side with respect to ' \(x\) ', we are…
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