Evaluate the value of the derivative given by \[\dfrac{d}{dx}\left( {{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}} \right)\]
(1) \[-\dfrac{{{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}}}{\cos e{{c}^{-1}}x}\left( \dfrac{-1}{x\sqrt{{{x}^{2}}-1}} \right){{\log }_{10}}3\]
(2) \[-\dfrac{{{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}}}{\cos e{{c}^{-1}}x}\left( \dfrac{-1}{\left| x \right|\sqrt{{{x}^{2}}-1}} \right){{\log }_{10}}3\]
(3) \[-\dfrac{{{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}}}{\cos e{{c}^{-1}}x}\left( \dfrac{-1}{x\sqrt{{{x}^{2}}-1}} \right){{\log }_{3}}10\]
(4) None of these

Hint: We are asked in the question to find the derivative of the given function. In order to find the derivative of the given function we will use the chain rule for the same. As we can see there are functions within functions. Hence, we will have the derivative of the given function.

Complete step by step solution:
According to the given question, we are given a function and we are asked to find the derivative of the function.
We will make use of the chain rule in order to find the derivative of the given function.
The given function we have is,
\[\dfrac{d}{dx}\left( {{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}} \right)\]
We know that the derivative of \[\dfrac{d}{dx}\left( {{a}^{x}} \right)={{a}^{x}} log {a}\], so we have the expression as,
\[\Rightarrow {{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}}.{{\log }_{10}}3.\dfrac{d}{dx}\left( {{\log }_{10}}\cos e{{c}^{-1}}x \right)\]
Now, applying the logarithm property on the inverse cosecant function, we have the expression as,
\[\Rightarrow {{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}}.{{\log }_{10}}3.\dfrac{1}{\cos e{{c}^{-1}}x}.\dfrac{d}{dx}\left( \cos e{{c}^{-1}}x \right)\]
Now, we will write the derivative of the inverse cosecant function and we have it as,
\[\Rightarrow {{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}}.{{\log }_{10}}3.\dfrac{1}{\cos e{{c}^{-1}}x}\left( \dfrac{-1}{\left| x \right|\sqrt{{{x}^{2}}-1}} \right)\]
The above expression can be much more reduced in size. Rearranging the above expression, we get the derivative as,
\[\Rightarrow -\dfrac{{{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}}{{\log }_{10}}3}{\cos e{{c}^{-1}}x}\left( \dfrac{1}{\left| x \right|\sqrt{{{x}^{2}}-1}} \right)\]
Therefore, the correct option is (2) \[-\dfrac{{{3}^{{{\log }_{10}}\cos e{{c}^{-1}}x}}}{\cos e{{c}^{-1}}x}\left( \dfrac{-1}{\left| x \right|\sqrt{{{x}^{2}}-1}} \right){{\log }_{10}}3\]

Note: The chain rule should be carefully applied on the function. Also, do not miss out any terms while differentiating the function. The derivative of certain functions should be known beforehand in order to avoid confusion and mistakes that could possibly make the answer wrong.