Question

Derivative of $\log {{\left( \log x \right)}^{2}}$ with respect to x is _ _ _ _ _ _
A. $\dfrac{2}{x\log x}$
B. $\dfrac{1}{x\log x}$
C. $\dfrac{1}{x\log {{x}^{2}}}$
D. $\dfrac{2}{\log x}$

Answer 1

Hint: This type of question is based on differentiation of logarithm function. Firstly we check whether any logarithm laws are to be used to simplify the question before solving it. Then using differentiation of logarithm expression which is $\dfrac{d\left( \log x \right)}{dx}=\dfrac{1}{x}\times \dfrac{d\left( x \right)}{dx}$ we find derivative of the expression and simplify it further to get the desired answer.

Complete step by step solution:
We have to find the derivative of $\log {{\left( \log x \right)}^{2}}$ with respect to x by using the differentiation formula of logarithm.
So as we know that differentiation of logarithm function is done as follows:
$\dfrac{d\left( \log x \right)}{dx}=\dfrac{1}{x}\times \dfrac{d\left( x \right)}{dx}$………$\left( 1 \right)$
To find derivative of $\log {{\left( \log x \right)}^{2}}$ we will use formula (1) as follows:
$\dfrac{d\left( \log {{\left( \log x \right)}^{2}} \right)}{dx}=\dfrac{1}{{{\left( \log x \right)}^{2}}}\times \dfrac{d\left( {{\left( \log x \right)}^{2}} \right)}{dx}$
Using formula (1) again on the right hand side and simplifying it we get,
$\begin{align}
  & \Rightarrow \dfrac{d\left( \log {{\left( \log x \right)}^{2}} \right)}{dx}=\dfrac{1}{{{\left( \log x \right)}^{2}}}\times 2\log x\times \dfrac{d}{dx}\left( \log x \right) \\
 & \Rightarrow \dfrac{d\left( \log {{\left( \log x \right)}^{2}} \right)}{dx}=\dfrac{1}{{{\left( \log x \right)}^{2}}}\times 2\log x\times \dfrac{1}{x} \\
 & \Rightarrow \dfrac{d\left( \log {{\left( \log x \right)}^{2}} \right)}{dx}=\dfrac{1}{{{\left( \log x \right)}^{2-1}}}\times 2\times \dfrac{1}{x} \\
 & \therefore \dfrac{d\left( \log {{\left( \log x \right)}^{2}} \right)}{dx}=\dfrac{2}{x\log x} \\
\end{align}$
So derivative of $\log {{\left( \log x \right)}^{2}}$ with respect to x is $\dfrac{2}{x\log x}$

So, the correct answer is “Option A”.

Note: Differentiation is a method used to find derivative of a function or we can say that the process of finding a derivative is known as differentiation. It is used to find the instantaneous rate of change in function depending on one of its variables. If the base of the logarithm function is not stated we consider it a natural logarithm function whose base is constant $e$ and use the formula stated above. The differentiation of logarithm function is different from the differentiation of exponential function as exponential function on differentiating gives the same expression with differentiation of the power multiplied. In some cases we use logarithm laws to simplify our expression before solving it.