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Last updated date: 08th Dec 2023
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MVSAT Dec 2023

The differential coefficient O \[f(\log x)\] w.r.t. \[x\] , where \[f(x) = \log x\] is
A. \[\dfrac{x}{{\log x}}\]
B. \[\dfrac{{\log x}}{x}\]
C. \[{(x\log x)^{ - 1}}\]
D.None of these.

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Hint: In this problem, we have to find the differential coefficient of the function. First, We need to find the value of the function \[f(\log x)\] by putting into the given function \[f(x) = \log x\] . then we will differentiate the resulting value of \[f(\log x)\] with respect to \[x\] .
We will use the formula \[\dfrac{{dy}}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}\] to get the required solution.

Complete step-by-step answer:
The differentiation of the process of finding the rate of change of a given function. This rate of change is called derivative of the function . It is denoted by \[\dfrac{{dy}}{{dx}}\] which in simple terms means rate of change of \[y\] with respect to \[x\] .
In order to determine the given function is \[f(x) = \log x\] .
First we find the value of \[f(\log x)\] by putting into the values of \[x\] . we get,
Let us assume, \[y = f(\log x) = \log (\log x)\] .
Using the formula , \[\dfrac{{dy}}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}\] we have ,
 \[\dfrac{{dy}}{{dx}}\left( {\log (\log x)} \right) = \dfrac{1}{{\log x}}\] . since, \[x = \log x\]
Now we have to differentiate the above equation with respect to \[x\] .
 \[ = \dfrac{1}{{\log x}}\dfrac{{\partial (\log x)}}{{dx}}\] .
The given value of \[y\] is function of function hence differentiating the inner function we have,
 \[\dfrac{{dy}}{{dx}} = \dfrac{\operatorname{l} }{{\log x}}\dfrac{{\partial (\log x)}}{{dx}} = \dfrac{1}{{\log x}}\left( {\dfrac{1}{x}} \right) = \dfrac{1}{{x\log x}}\] .
The above result can also be written as
Hence, The differential coefficient O \[f(\log x)\] w.r.t. \[x\] , where \[f(x) = \log x\] is
 \[\dfrac{{dy}}{{dx}} = = \dfrac{1}{{x\log x}} = {\left( {x\log x} \right)^{ - 1}}\] .
Hence option C is the correct answer.
So, the correct answer is “Option C”.

Note: We use differentiation to find the rate of change of function .
We use formulas for finding the derivative of the function.
For example \[\dfrac{{dy}}{{dx}}({x^n}) = n{x^{n - 1}}\] , \[\dfrac{{dy}}{{dx}}(\log x) = \dfrac{1}{x}\]
Differentiation of constant is 0.
Differentiation of \[x\] with respect to \[x\] is 1. i.e. \[\dfrac{{dy}}{{dx}}(x) = 1\]
For differentiation of function of function we use chain rule , i.e. we first differentiate the given function and then differentiate the inner function .
For example , let \[y = \log \left( {\log x} \right)\]
Then \[\dfrac{{dy}}{{dx}} = \dfrac{\partial }{{dx}}(\log \log x) = \dfrac{1}{{\log x}} \times \dfrac{\partial }{{dx}}\log x\]
Hence \[\dfrac{{dy}}{{dx}} = \dfrac{1}{{x\log x}}\] .
Similarly we can differentiate function of function.