Question

#Quality Question
@Logrithms

If$\bf y  = [log \{log(logx) \}] {}^{2}$

then Find out the value of dy/dx.
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Answer 1

《¤¤¤¤¤¤¤¤¤¤¤¤¤¤》

▪Given :-

$\bf y  = [log \{log(logx) \}] {}^{2}$

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▪To Calculate :-

$\bf \large \blue{{dy/dx}}$

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▪Formulae Used :-

$\bigstar \:  \bf  \frac{d}{dx} f(x) {}^{2}  = 2f(x) \frac{d}{dx} f(x) \\  \\   \bigstar   \bf\frac{d}{dx} log(f(x)) =  \frac{1}{f(x)} . \frac{d}{dx} f(x)$

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▪Solution :-

$\bf y  = [log \{log(logx) \}] {}^{2}$

Differentiating both side w.r.t x

$\bf\frac{dy}{dx}  =  \frac{d}{dx} [log \{log(logx) \}] {}^{2}  \\  \\  = \sf2[log \{log(logx) \}] .\frac{d}{dx}  [log \{log(logx) \}] \\  \\  = \sf 2[log \{log(logx) \}]  \times   \frac{1}{ \{log(log \: x) \} } \\  \times     \sf\frac{ d}{dx}   \{log{(log \: x)} \} \\  \\   = \sf 2[log \{log(logx) \}]  \times   \frac{1}{ \{log(log \: x) \} } \\  \times   \sf\frac{1}{log \: x}  \times \frac{d}{dx}log \: x \\  \\  = \sf 2[log \{log(logx) \}]  \times   \frac{1}{ \{log(log \: x) \} } \\  \times   \sf\frac{1}{log \: x}  \times  \frac{1}{x}\\\\{ \underline{\boxed{\bf \frac{dy}{dx}=\frac{2[log \{log(logx)\}]}{x log \: x.\{log(log \: x)\} }}}}$

$\Large \red{\mathfrak{  \text{W}hich \:   \: is  \:  \: the  \:  \: required }}\\ \huge \red{\mathfrak{ \text{ A}nswer.}}$

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Answer 2

Given that , y = [ log { log ( log x ) } ]² .

Exigency To Find : The value of dy / dx .

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

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

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

$\qquad \sf \:(\:II\:)\: \dfrac{d}{dx}\:log\:\big[ f (x) \big] \:\:=\: \dfrac{1}{\:f\:(\:x\:) } \:\times \:f'\:(\:x\:) \:$

$\rule{150}{1.5}$

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

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

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

⠀⠀⠀⠀⠀We get ,

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