# y= log (log 8 (log 7 (log 6 (log 5 (x))))) find the domain of dy/dx

y= log (log 8 (log 7 (log 6 (log 5 (x))))) find the domain of dy/dx

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𝑦 = xxxxxxxxxxxxxxxxxxxxxxxx 𝑥)))) y xx the xxxxxxxxx xxx xx xxxx xx xxxx its derivative: 𝑑𝑦

𝑑𝑥

𝑑𝑦

𝑑𝑥 = 𝑦′ x xxxxxxxxxxxxxxxxxxxx 𝑥))) ∙ xx x xxx xxxxxxxxxxxxxxxxxxxx 𝑥xxxx′ x xxxxxxxxxxxxxxxxxxxx 𝑥))) xxx xx x xxx xxxxxxxxxxxxxxx 𝑥)) xxx ln x xxxxxxxxxxxxxxx 𝑥xxx′ x xxxxxxxxxxxxxxxxxxxx 𝑥xxx xxx ln x xxx 1log7(log6(log5 𝑥)) ∙ xx 8 ∙ 1log6(log5 𝑥) ∙ ln x [log6(log5 𝑥)]′ x xxxxxxxxxxxxxxxxxxxx 𝑥xxx ∙ xx 9 ∙ xxxxxxxxxxxxxxx 𝑥)) ∙ xx x xxx 1log6(log5 𝑥x ∙ xx xx

xxxxxxxxx 𝑥 ∙ ln 6 xxxxx 𝑥]′ x xxxxxxxxxxxxxxxxxxxx 𝑥xxx xxx xx x ∙ 1log7(log6(log5 𝑥xx ∙ xx x ∙ 1log6(log5 𝑥x ∙ ln 7 ∙ xxxxx 𝑥 ∙ xx xx

∙ 1

𝑥 xxx ln 5 xxx xx order to find xxx xxxxxx of xxxx xxxxxxxxxx xx have xxxxxxxx xxxxxxxxxxx Firslty all xxxxxxxxx of xxxxxxxxxx must xx greater

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x xx the function. Now we xxxx xx find its derivative:

Now xx xxxxx xx find xxx xxxxxx of xxxx xxxxxxxxxx xx have xxxxxxxx conditions. Firslty all xxxxxxxxx of xxxxxxxxxx xxxx xx xxxxxxx xxxx zero:

Secondly, all xxxxxxxxxxxx must xx different xxxx zero.

xxxxxxxxx , x x , x are xxxxxx real xxx different xxxx zero.

The final domain of xxx xxxxxxxxx is xxx xxxxxxxxx xx xxx conditions. The xxxxxx xxxxxxxx xxx the xxxxx 4 xxxxxxxxxx xxx

Because xx greater than xxx other numbers xx conditions. In xxxxx words if the x must be xxxxxxx than xx 1, xx than it must xx greater than .

We xxxx exclude xxxx the xxxxxx xxx values of x xxxx xxxxxxxxxx 6-10. Since numbers 0, 1, 5, x xxx xxx xx the interval, xxxxx is xx xxxxx xx xxxxxxxxx xxxxx The xxxx number we xxxx exclude xx :

This is xxx xxxxx solution.

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𝑦 x log(log8(log7(log6(log5 𝑥xxxx Y xx the function. Now xx have to xxxx xxx derivative: 𝑑𝑦

𝑑𝑥

𝑑𝑦

𝑑𝑥 = 𝑦′ x xxxxxxxxxxxxxxxxxxxx 𝑥))) xxx xx 10 xxx [log8(log7(log6(log5 𝑥)))]′ = xxxxxxxxxxxxxxxxxxxx 𝑥xxx xxx xx xx xxx 1log7(log6(log5 𝑥xx xxx xx 8 xxxxxxxxxxxxxxx 𝑥xxx′ x xxxxxxxxxxxxxxxxxxxx 𝑥xxx xxx xx xx xxx xxxxxxxxxxxxxxx 𝑥)) xxx xx x ∙ 1log6(log5 𝑥x xxx xx x [log6(log5 𝑥)]′ x xxxxxxxxxxxxxxxxxxxx 𝑥))) ∙ ln xx xxx xxxxxxxxxxxxxxx 𝑥)) xxx ln x xxx 1log6(log5 𝑥x ∙ xx xx

∙ 1log5 𝑥 ∙ xx 6 xxxxx 𝑥]′ = 1log8(log7(log6(log5 𝑥xxx ∙ ln 10 xxx xxxxxxxxxxxxxxx 𝑥)) xxx ln x xxx 1log6(log5 𝑥) xxx ln x xxx xxxxx 𝑥 ∙ ln xx

∙ 1

𝑥 ∙ ln 5 xxx in xxxxx xx find the domain of xxxx xxxxxxxxxx xx xxxx xxxxxxxx xxxxxxxxxxx Firslty all arguments of xxxxxxxxxx must be

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## domain, range and plot of the function in the attached file. All steps shown.

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All xxxxx xxxxxx

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xxxxxx

xxxxx

Domain:

xxxxxx

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## answer

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xxxxxx we xxxxxx find xxxxx x is defined.

xxxx that the xxxxxx of xxx log xxxxxxxx xx always the xxxxxxxx xxxx xxxxxxx xxxxxxx we xxx xxxxxxx xxxx xxxxxxx xxxxxxxxxx -- xxxxx xx xxx not).

xxxxxxxxxxxxxxxxxxx

Log7log6log5x>log8=1

Log6log5x>7(8)(0)=7

xxxxxxxx 7)(8)(0)=6(7)

xxxxxxxxxxxxxxxxxxxxxxx

Hence,

xxxxxxxxx xxx value of x such that xx in xxx domain. Since xxx x is a differentiable function, log(log(x)) is xxxxxxxxxxxxxxx xxx so xxx The domain xx dy/dx xx xxx same xx xxx xxxxxx of y.

xxxxxxxx in xxxxxxx are raise to xxx xxxxx to xxx first

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