Question

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

Answer 1

Hint: Differentiation of logarithmic functions are mainly based on the chain rule. We can generalize any differentiable function with a logarithmic function. Differentiation of any constant is zero. Differentiation of a constant and a function is equal to constant times the differentiation of the function. The differentiation of log is only under the base \[e\] but we can differentiate under other bases. Derivative of \[\log x\] is given as \[\dfrac{d}{{dx}}\log x = \dfrac{1}{x}\].

Complete step-by-step solution:
It is given in the question that \[f(x) = \log x\].
So, we have to find the composite function \[f(\log x)\] first and then differentiate it with respect to x.
So, we substitute the value of x as \[\log x\] in the function. So, we get,
\[ \Rightarrow f(\log x) = \log \log x\]
Now, we have to differentiate this composite function with respect to x.
Now, Let us assume $u = \log x$. So substituting $\log x$ as $u$, we get,
\[ \Rightarrow f(\log x) = \log \left( u \right)\]
Differentiating both sides with respect to x, we get,
\[ \Rightarrow \dfrac{d}{{dx}}\left[ {f(\log x)} \right] = \dfrac{d}{{dx}}\left[ {\log \left( u \right)} \right]\]
We know that the derivative of logarithmic function \[(\log x)\] with respect to x is \[\dfrac{1}{x}\].
\[ \Rightarrow \dfrac{d}{{dx}}\left[ {f(\log x)} \right] = \left( {\dfrac{1}{u}} \right)\dfrac{{du}}{{dx}}\]
Now, substituting back u as $\log x$, we get,
\[ \Rightarrow \dfrac{d}{{dx}}\left[ {f(\log x)} \right] = \left( {\dfrac{1}{{\log x}}} \right)\dfrac{{d\left( {\log x} \right)}}{{dx}}\]
\[ \Rightarrow \dfrac{d}{{dx}}\left[ {f(\log x)} \right] = \left( {\dfrac{1}{{\log x}}} \right)\left( {\dfrac{1}{x}} \right)\]
So, simplifying the expression, we get,
\[ \Rightarrow \dfrac{d}{{dx}}\left[ {f(\log x)} \right] = \left( {\dfrac{1}{{x\log x}}} \right)\]
This can further be written as \[f'(\log x) = {(x\log x)^{ - 1}}\]
Hence, option B is the correct answer.

Note: The slope is the rate of change of \[y\] with respect to \[x\] that means if \[x\] is increased by an additional unit the change in \[y\] is given by \[\dfrac{{dy}}{{dx}}\]. Let us understand with an example, the rate of change of displacement of an object is defined as the velocity \[\dfrac{{Km}}{{hr}}\] that means when time is increased by one hour the displacement changes by a kilometre. For solving derivative problems different techniques of differentiation must be known thoroughly.