A) \[\frac{{{2}^{x-1}}\left\{ -2x\,\text{cos}\text{e}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{{{x}^{3/2}}}\]
B) \[\frac{{{2}^{x-1}}\left\{ -2x\cos \text{e}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{x}\]
C) \[\frac{2x\left\{ -2x\text{cose}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{{{x}^{\text{3/2}}}}\]
D) None of these
Correct Answer: A
Solution :
\[\frac{dA}{dx}=\frac{\sqrt{x}\{{{2}^{x}}{{\log }_{e}}2\cot x-{{2}^{x}}\text{cose}{{\text{c}}^{2}}x\}-{{2}^{x}}\cot x\frac{1}{2\sqrt{x}}}{x}\] \[=\frac{{{2}^{x-1}}\left\{ -2x\,\text{cose}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{{{x}^{3/2}}}\].
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