AP EAMCET · Maths · Differentiation
At \(x=\frac{\pi^2}{4}, \frac{d}{d x}\left(\tan ^{-1}(\cos \sqrt{x})+\sec ^{-1}\left(e^x\right)\right)=\)
- A \(\frac{1}{\sqrt{e^{\frac{\pi^2}{2}}-1}}-\frac{1}{\pi}\)
- B \(\frac{\pi}{4}+\frac{1}{\sqrt{\mathrm{e}^{\pi^2}+\mathrm{e}^{\pi^2 / 2}}}\)
- C \(\frac{1}{\sqrt{\mathrm{e}^{\pi^2}+\mathrm{e}^{\pi^2 / 2}}}+\frac{2}{\pi} \cot \left(\frac{\sqrt{\pi}}{2}\right)\)
- D \(\frac{1}{\sqrt{\mathrm{e}^\pi}}+\frac{1}{\pi}\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{\sqrt{e^{\frac{\pi^2}{2}}-1}}-\frac{1}{\pi}\)
Step-by-step Solution
Detailed explanation
\[ \begin{aligned} & \text { } \frac{d}{d x}\left(\tan ^{-1}(\cos \sqrt{x})+\sec ^{-1}\left(\mathrm{e}^x\right)\right) \\ & =\frac{(-\sin \sqrt{x})}{1+\cos ^2 \sqrt{x}} \cdot\left(\frac{1}{2 \sqrt{x}}\right)+\frac{1}{e^x \sqrt{e^{2 x}-1}} \cdot e^x \end{aligned} \] but when…
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