Question

If y = [log(log(logx))]? , find dy/dx​

Answer 1

Answer:

wish it is helpful to you

Answer 2

Given that ,

• y = [ log { log ( log x ) } ]² .

Exigency To Find :

• The value of dy / dx .

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\begin{gathered}\qquad \:\: \underline {\pmb{\pink{\cal { FORMULAS \:\:USED \:\: IN \:\: CALCUTION \:\::\:}}}}\\\\\end{gathered}$

$\begin{gathered}\qquad \sf \:(\:I\:)\: \dfrac{d}{dx}\:f(x)^2 \:\:=\: 2\:f\:(\:x\:) \:\times \: \dfrac{d}{dx}\:f'\:(\:x\:) \:\\\\\end{gathered}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀¤ Finding Value of dy / dx :

$\begin{gathered}\qquad \dashrightarrow \sf y\:=\:\bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg]^2 \:\:\\\\\end{gathered}$

$\begin{gathered}\qquad \bigstar \:\underline {\purple {\sf By \:,\:Differentiating \:\:both \:\:sides\:\: w.r.t.x \:, \:}}\\\end{gathered}$

⠀⠀⠀⠀⠀We get ,

$\begin{gathered} :\implies \sf \dfrac{dy}{dx} \:=\: \dfrac{d}{dx} \:\bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg]^2 \:\: \:\\\\\\:\implies \sf \dfrac{dy}{dx} \:=\: \:2 \bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg] \:\times \:\dfrac{d}{dx}\bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg] \:\: \:\\\\\\:\implies \sf \dfrac{dy}{dx} \:=\: \:2 \bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg] \:\times \:\dfrac{1}{log(log \:x)}\:\times \dfrac{d}{dx} \: log\:\big\{ log ( log \:x\:)\:\big\} \:\: \:\\\\\\:\implies \sf \dfrac{dy}{dx} \:=\: \:2 \bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg] \:\times \:\dfrac{1}{log(log \:x)}\:\times \dfrac{1}{log\:x} \times \dfrac{d}{dx} \: log\: ( log \:x\:)\: \:\: \:\\\\\\:\implies \sf \dfrac{dy}{dx} \:=\: \:2 \bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg] \:\times \:\dfrac{1}{log(log \:x)}\:\times \dfrac{1}{log\:x} \times \dfrac{1}{x} \: \: \:\: \:\\\\\\:\implies \sf \dfrac{dy}{dx} \:=\: \dfrac{\:2 \bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg]}{x(log\:x)\: \big[ log ( log \:x \:)\:\big] } \:\: \:\\\\\\:\implies \underline{\boxed {\pmb{\frak{ \purple { \dfrac{dy}{dx} \:=\: \dfrac{\:2 \bigg[ \: log\:\big\{ log ( log \:x\:)\:\big\} \:\:\bigg]}{x(log\:x)\: \big[ log ( log \:x \:)\:\big] } }}}}} \:\: \:\\\\\end{gathered}$