Question

What is the derivative of $ln\left( ln\left( {{x}^{2}} \right) \right)$?

Answer 1

Hint: We need to find the derivative of the function given. Now, this is not a general function, it is a complex one. So we apply various rules of derivatives here such as the chain rule. Using these we will perform the derivative of the whole function. We differentiate functions one by one and then multiply them to obtain the result. Formulas such as the derivative of logarithm or derivative of polynomials should be known.

Complete step by step solution:
We have the function composed into a function, so we simply convert each of the sub parts as a function and we will take the derivative of the innermost function, then the outer one and so on. We will denote all the functions as following:
$f(x)=ln(g(x))$
$g(x)=ln(h(x))$
$h(x)={{x}^{2}}$
Now, we can differentiate these one by one and we get:
$\dfrac{df}{dg}=\dfrac{1}{g(x)}$
$\dfrac{dg}{dh}=\dfrac{1}{h(x)}$
$\dfrac{dh}{dx}=2x$
We have used the following formulae here:
$\dfrac{d\ln (a(x))}{da}=\dfrac{1}{a(x)}$
$\dfrac{d{{x}^{2}}}{dx}=2x$
Now, we need to apply chain rule to the given function:
$\dfrac{df}{dx}=\dfrac{df}{dg}\times \dfrac{dg}{dh}\times \dfrac{dh}{dx}$
We simply plug in the values we have calculated above, so we obtain:
$\dfrac{df}{dx}=\dfrac{1}{g(x)}\times \dfrac{1}{h(x)}\times 2x$
Putting the values of the functions $g,h$ we get:
$\dfrac{df}{dx}=\dfrac{1}{\ln (h(x))}\times \dfrac{1}{{{x}^{2}}}\times 2x$
$\Rightarrow \dfrac{df}{dx}=\dfrac{1}{\ln ({{x}^{2}})}\times \dfrac{2}{{{x}^{{}}}}$

So, we get the result as following:
$\dfrac{df}{dx}=\dfrac{2}{x\ln ({{x}^{2}})}$
Hence, we have obtained the derivative of the function.

Note:
While differentiating, do not integrate by mistake. It is often common to interchange the formulae of both leading to an incorrect result. Moreover, while applying chain rule, do not forget to differentiate till the last term because that term often gets missed. Try to remember as much formulae as possible so that you don’t waste much time while doing the question.