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Find the derivative of the following function. \[y={{\log }_{3}}x-{{\log }^{-1}}{{3}^{-1}}\times \log {{x}^{-1}}\]

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Hint: To find the derivative of the function given in the question, one must start by
 simplifying the given function using properties of logarithmic function and then differentiating the terms given in the function using sum and product rule of differentiation.

To find the derivative of the function\[y={{\log }_{3}}x-{{\log }^{-1}}{{3}^{-1}}\times \log {{x}^{-1}}\], we will differentiate it with respect to the variable\[x\]using some logarithmic properties.
We will first simplify the given function.
We know that\[{{\log }_{b}}a=\dfrac{\log a}{\log b}\].
Substituting\[a=x,b=3\]in the above equation, we get\[{{\log }_{3}}x=\dfrac{\log x}{\log 3}\]. \[...(1)\]
We also know that\[{{\log }^{-1}}a=\dfrac{1}{\log a}\]. \[...(2)\]
  Substituting\[a={{3}^{-1}}\], we get\[{{\log }^{-1}}{{3}^{-1}}=\dfrac{1}{\log {{3}^{-1}}}\].
We know that\[\log {{a}^{-1}}=-\log a\].
Thus, we have\[\dfrac{1}{\log {{3}^{-1}}}=\dfrac{1}{-\log 3}\].
Merging the above equation with equation\[(3)\], we get\[{{\log }^{-1}}{{3}^{-1}}=\dfrac{1}{\log {{3}^{-1}}}=\dfrac{1}{-\log 3}\]. \[...(4)\]
Substituting\[a=x\]in equation\[(2)\], we get\[\log {{x}^{-1}}=-\log x\]. \[...(5)\]
Substituting equation\[(1)\], \[(4)\]and\[(5)\]in the given equation\[y={{\log }_{3}}x-{{\log }^{-1}}{{3}^{-1}}\times \log {{x}^{-1}}\], we get\[y={{\log }_{3}}x-{{\log }^{-1}}{{3}^{-1}}\times
 \log {{x}^{-1}}=\dfrac{\log x}{\log 3}-\left( \dfrac{1}{-\log 3} \right)-\log x\].
Simplifying the above equation, we get\[y=\dfrac{\log x}{\log 3}-\dfrac{\log x}{\log 3}=0\].
We know that the differentiation of a constant function with respect to any variable is 0.
Thus, we have\[\dfrac{dy}{dx}=0\]as\[y=0\].
We get the derivative of the given function 0 as by simplifying the function, we get a
 constant function and we know that the derivative of a constant function is zero.
The first derivative of any function signifies the slope of the function. As the graph of
constant function is a horizontal line, its slope is zero, which justifies that the value of the
 first derivative of the given function is 0.
Note: We can also solve the given question by actually differentiating each term of the
 function using sum and product rule of differentiation. However, it will be time consuming.
 Also, it’s better to simplify the function before finding its derivative as it’s easy to
 differentiate a simple function instead of a complicated one.
Last updated date: 26th Sep 2023
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