Limits Continuity And Differentiability Question 90
Question: If $ y=log _{10}x+log _{x}10+log _{x}x+log _{10}10 $ then what is $ {{( \frac{dy}{dx} )} _{x=10}} $ equal to?
Options:
A) 10
B) 2
C) 1
D) 0
Show Answer
Answer:
Correct Answer: D
Solution:
$ y={\log _{10}}x+{\log _{x}}10+{\log _{x}}x+{\log _{10}}10 $
$ y={\log _{10}}x+{\log _{x}}10+1+1 $ Differentiating equation w.r.t.x $ \frac{dy}{dx}=\frac{1}{x{\log _{e}}10}-\frac{1}{{{({\log _{10}}x)}^{2}}}.\frac{1}{(x\log 10)} $
$ =\frac{1}{x{\log _{e}}10}[ 1-\frac{1}{{{({\log _{10}}x)}^{2}}} ] $
$ {{( \frac{dy}{dx} )} _{x=10}}=\frac{1}{10\log _{e}10}[1-1]=0 $
$ [ Note:{\log _{x}}10=\frac{{\log _{10}}10}{{\log _{10}}x}=\frac{1}{{\log _{10}}x} \\ \frac{d}{dx}[ \frac{1}{{\log _{10}}x} ]=-{{({\log _{10}}x)}^{-2}}\times \frac{1}{x{\log _{e}}10} \\ =-\frac{1}{{{({\log _{10}}x)}^{2}}x{\log _{e}}10} ] $
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