If y=( x)^{( x)^{( x)^{·s^{∞}}}} then dy/dx is equal to

Correct Answer is : −y2tan⁡x1−yln⁡(cos⁡x)-y² x/1-y ( x)−1−yln(cosx)y2tanx

Correct Answer is : −y2tan⁡x1−yln⁡(cos⁡x)-y² x/1-y ( x)−1−yln(cosx)y2tanx

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UPSC NDA Math Model Paper 3

Question : 23 of 120

Marks: +1, -0

y=(\cos x)^{(\cos x)^{(\cos x)^{\cdots^{\infty}}}}

\frac{dy}{dx}

is equal to

Solution:

Given:

y=(\cos x)^{(\cos x)^{(\cos x)^{\cdots^{\infty}}}}

To find:

\frac{dy}{dx}

y=(\cos x)^{(\cos x)^{(\cos x)^{\cdots^{\infty}}}}

\Rightarrow y=(\cos x)^y

Taking

\ln

of both sides,

\Rightarrow \ln y=\ln (\cos x)^y

\Rightarrow \ln y=y \times \ln (\cos x)(\text{Using} \ln M^{p}

=p \times \ln M)

Differentiating on both sides w.r.t x

\Rightarrow \frac{1}{y} \times \frac{dy}{dx}=\frac{y}{\cos x}(\sin x)+\frac{dy}{dx} \times \ln (\cos x)

\Rightarrow \frac{dy}{dx}\left(\frac{1}{y}-\ln (\cos x)\right)=-y \tan x

\Rightarrow \frac{dy}{dx}\left(\frac{1-y \ln (\cos x)}{y}\right)=-y \tan x

\Rightarrow \frac{dy}{dx}=-\frac{y^2 \tan x}{1-y \ln (\cos x)}

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