Question

What is derivative of $y = \ln \left( {\ln \left( x \right)} \right)$ with respect to $x$ ?

Answer 1

Hint : In the given problem, we are required to differentiate $y = \ln \left( {\ln \left( x \right)} \right)$ with respect to x. Since, $y = \ln \left( {\ln \left( x \right)} \right)$ is a composite function, so we will have to apply chain rule of differentiation in the process of differentiating $y = \ln \left( {\ln \left( x \right)} \right)$ . So, differentiation of $y = \ln \left( {\ln \left( x \right)} \right)$ with respect to x will be done layer by layer using the chain rule of differentiation. Also the derivative of $\ln \left( x \right)$ with respect to $x$ must be remembered in order to solve the given question.

Complete step by step solution:
To find derivative of $y = \ln \left( {\ln \left( x \right)} \right)$ with respect to $x$, we have to find differentiate $y = \ln \left( {\ln \left( x \right)} \right)$ with respect to $x$ . So, Derivative of $y = \ln \left( {\ln \left( x \right)} \right)$ with respect to $x$ can be calculated as $\dfrac{d}{{dx}}\left[ {\ln \left( {\ln \left( x \right)} \right)} \right]$ .
Now, $\dfrac{d}{{dx}}\left[ {\ln \left( {\ln \left( x \right)} \right)} \right]$
Now, Let us assume $u = \ln x$ . So substituting $\ln x$ as $u$, we get,
$=$ $\dfrac{d}{{dx}}\left[ {\ln \left( u \right)} \right]$
Now, we know that the derivative of $\ln x$ with respect to x is $\left( {\dfrac{1}{x}} \right)$ . So, we get,
$=$ $\left( {\dfrac{1}{u}} \right)\dfrac{{du}}{{dx}}$
Now, putting back $u$ as $\ln x$, we get,
$=$ $\left( {\dfrac{1}{{\ln x}}} \right)\dfrac{{d\left( {\ln x} \right)}}{{dx}}$ because $\dfrac{{du}}{{dx}} = \dfrac{{d\left[ {\ln x} \right]}}{{dx}}$
Now, we know that the derivative of $\ln x$ with respect to x is $\left( {\dfrac{1}{x}} \right)$ . So, we get,
$=$ $\left( {\dfrac{1}{{\ln x}}} \right)\left( {\dfrac{1}{x}} \right)$
Simplifying the expression further, we get,
$=$ $\dfrac{1}{{x\ln x}}$
So, the derivative of $y = \ln \left( {\ln \left( x \right)} \right)$ with respect to $x$ is $\dfrac{1}{{x\ln x}}$ .
So, the correct answer is “ $\dfrac{1}{{x\ln x}}$ ”.

Note : The given problem may also be solved using the first principle of differentiation. The derivatives of basic functions must be learned by heart in order to find derivatives of complex composite functions using chain rule of differentiation. The chain rule of differentiation involves differentiating a composite by introducing new unknowns to ease the process and examine the behaviour of function layer by layer.