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

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.