A) \[a\cos a+{{a}^{2}}\sin a\]
B) \[a\sin a+{{a}^{2}}\cos a\]
C) \[2a\sin a+{{a}^{2}}\cos a\]
D) \[2a\cos a+{{a}^{2}}\sin a\]
Correct Answer: C
Solution :
\[\frac{d}{da}\,[{{a}^{2}}\sin a]=2a\sin a+{{a}^{2}}\cos a.\] Aliter : Apply L-Hospital?s rule, \[\underset{h\to 0}{\mathop{\lim }}\,\,\frac{{{(a+h)}^{2}}\sin (a+h)-{{a}^{2}}\sin a}{h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{2\,(a+h)\,\sin \,(a+h)+{{(a+h)}^{2}}\cos \,(a+h)}{1}\] \[=2a\,\,\sin a+{{a}^{2}}\cos \,\,a.\]
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