Question

Differentiation Jee. Differentiate log_2(log_3(log_5 (x)))

Answer 1

$\large\underline{\sf{Solution-}}$

The given function is

$\rm :\longmapsto\: log_{2}( log_{3}( log_{5}(x) ) )$

Let we assume that

$\rm :\longmapsto\: y \: = \: log_{2}( log_{3}( log_{5}(x) ) )$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx} y \: = \dfrac{d}{dx}\: log_{2}( log_{3}( log_{5}(x) ) )$

We know,

$\red{\boxed{ \rm{ \: \: \dfrac{d}{dx} log_{a}(x) = \frac{1}{xloga} \: \: }}}$

So, using this identity, we get

$\rm :\longmapsto\:\dfrac{dy}{dx} = \dfrac{1}{ log_{3}( log_{5}(x) ) log2}\dfrac{d}{dx}log_{3}( log_{5}(x)$

$\rm :\longmapsto\:\dfrac{dy}{dx} = \dfrac{1}{ log_{3}( log_{5}(x) )log2} \times \dfrac{1}{ log_{5}(x)log3 } \dfrac{d}{dx}log_{5}(x)$

$\rm :\longmapsto\:\dfrac{dy}{dx} = \dfrac{1}{ log_{3}( log_{5}(x) )log2} \times \dfrac{1}{ log_{5}(x)log3 } \times \dfrac{1}{xlog5}$

Hence,

$\red{\boxed{ \rm{ \: \: \: \:\dfrac{dy}{dx} = \dfrac{1}{x log_{3}( log_{5}(x)) log_{5}(x) log2log3log5} \: \: \: }}}$

Additional Information :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$