Question

Diffrentiate $\sf \large [log \{log(logx) \}] {}^{2}$ w.r.t x
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Answer 1

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▪Question :-

Differentiate $[\bf log \{log(logx) \}] {}^{2}$w.r.t x

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▪Let :-

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

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▪To Calculate :-

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

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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

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

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

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Answer 2

$\huge\red{\underline{\underline{\bf Answer}}}$

▪Let :-

$\bf y = [log \{log(logx) \}] {}^{2}y =[log{log(logx)}] </p><p>2$

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▪To Calculate :-

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

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▪Formulae Used :-

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

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▪Solution :

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

Differentiating both side w.r.t x

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