d dx [ e^x ]= - Vidyadip

MCQ

${d \over {dx}}\left[ {\log \sqrt {\sin \sqrt {{e^x}} } } \right]=$

✓
${1 \over 4}{e^{x/2}}\cot ({e^{x/2}})$
B
${e^{x/2}}\cot ({e^{x/2}})$
C
${1 \over 4}{e^x}\cot \,({e^x})$
D
${1 \over 2}{e^{x/2}}\cot \,({e^{x/2}})$

✓

Answer

Correct option: A.

${1 \over 4}{e^{x/2}}\cot ({e^{x/2}})$

a
(a) $\frac{d}{{dx}}[\log \sqrt {\sin \sqrt {{e^x}} } ] = \frac{d}{{dx}}\left[ {\frac{1}{2}\log (\sin \sqrt {{e^x}} )} \right]$

$ = \frac{1}{2}\cot \sqrt {{e^x}} \frac{1}{{2\sqrt {{e^x}} }}{e^x} = \frac{1}{4}{e^{x/2}}\cot ({e^{x/2}})$

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