Differentiation Question 375
Question: If $ y=x^{3}\log {\log _{e}}(1+x) $ , then $ {y}’’(0) $ equals
[AMU 1999]
Options:
A) 0
B) - 1
C) $ 6\log { _{e}}2 $
D) 6
Show Answer
Answer:
Correct Answer: A
Solution:
$ y=x^{3}\log {\log _{e}}(1+x) $
Therefore $ {y}’=3x^{2}\log {\log _{e}}(1+x)+\frac{x^{3}}{1+x}.\frac{1}{{\log _{e}}(1+x)} $
Therefore $ {y}’’=6x\log {\log _{e}}(1+x)+\frac{3x^{2}}{{\log _{e}}(1+x)}.\frac{1}{(1+x)} $
$ -\frac{x^{3}}{{{(1+x)}^{2}}{\log _{e}}(1+x)}-\frac{x^{3}}{{{(1+x)}^{2}}}.\frac{1}{{{[{\log _{e}}(1+x)]}^{2}}}+\frac{3x^{2}}{(1+x){\log _{e}}(1+x)} $
Therefore $ {y}’’(0)=0 $ .
BETA
