Question

Find the derivative of log[log(logx⅝)]​

Answer 1

Question :

Find the derivative of

$\sf log( log( logx {}^{ \frac{5}{8} } ) )$

Theory :

Chain rule

Let y = f(t) ,t= g(u) and u = m(x) ,then

$\sf \dfrac{dy}{dx} = \dfrac{dy}{dt} \times \dfrac{dt}{du} \times \dfrac{du}{dx}$

Solution :

$\sf \: y = log( log( log \: x {}^{ \frac{5}{8} } ) )$

Now Differentiate with respect to x by using chain rule .

$\sf \frac{dy}{dx} = \frac{d \: log( log( log \: x {}^{ \frac{5}{8} } ) ) }{d \: log( log \: x {}^{ \frac{5}{8} } ) } \times \frac{d \: log( log \: x {}^{ \frac{5}{8} } ) }{d \: log \: x {}^{ \frac{5}{8} } } \times \frac{d \: log \: x {}^{ \frac{5}{8} } }{dx}$

$\sf\implies \frac{dy}{dx} = \frac{1}{ log( log \: x {}^{ \frac{5}{8} } ) } \times \frac{1}{ log(x {}^{ \frac{5}{8} } ) } \times \frac{5}{8} \frac{d \: log(x) }{dx}$

$\sf \implies \dfrac{dy}{dx} = \dfrac{5}{8 \times log( log \: x {}^{ \frac{5}{8} } ) \times log \: x {}^{ \frac{5}{8} } } \times \dfrac{1}{x}$

$\sf \implies \dfrac{dy}{dx} = \dfrac{5}{8} \times \dfrac{1}{x log( log \: x {}^{ \frac{5}{8} } ) \times log(x {}^{ \frac{5}{8} } ) }$

It is the required solution

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Formula's of differentiation:

$\sf1) \frac{d \log \: x}{dx} = \frac{1}{x}$

$\sf2) \frac{d(constant)}{dx} = 0$

Properties of Logarithm:

$\sf log(x {}^{n} ) =n \log \: x$