A) 10
B) 2
C) 1
D) 0
Correct Answer: D
Solution :
[d] \[y={{\log }_{10}}x+{{\log }_{x}}10+{{\log }_{x}}x+{{\log }_{10}}10\] \[y={{\log }_{10}}x+{{\log }_{x}}10+1+1\] Differentiating equation w.r.t.x \[\frac{dy}{dx}=\frac{1}{x{{\log }_{e}}10}-\frac{1}{{{({{\log }_{10}}x)}^{2}}}.\frac{1}{(x\log 10)}\] \[=\frac{1}{x{{\log }_{e}}10}\left[ 1-\frac{1}{{{({{\log }_{10}}x)}^{2}}} \right]\] \[{{\left( \frac{dy}{dx} \right)}_{x=10}}=\frac{1}{10\,{{\log }_{e}}10}[1-1]=0\] \[\left[ \begin{align} & Note:{{\log }_{x}}10=\frac{{{\log }_{10}}10}{{{\log }_{10}}x}=\frac{1}{{{\log }_{10}}x} \\ & \frac{d}{dx}\left[ \frac{1}{{{\log }_{10}}x} \right]=-{{({{\log }_{10}}x)}^{-2}}\times \frac{1}{x{{\log }_{e}}10} \\ & =-\frac{1}{{{({{\log }_{10}}x)}^{2}}x{{\log }_{e}}10} \\ \end{align} \right]\]
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