if-y-sinx-sinx-sinx-find-dy-dx-

Question Number 25567 by rita1608 last updated on 11/Dec/17

i f y {(s i n x)}^{{(s i n x)}^{(s i n x) .^{.^{.^{. \infty}}}}}

f i n d d y / d x .

Commented by prakash jain last updated on 11/Dec/17

y = {(s i n x)}^{{(s i n x)}^{(s i n x) .^{.^{.^{. \infty}}}}} ?

Answered by prakash jain last updated on 11/Dec/17

y = {(s i n x)}^{{(s i n x)}^{(s i n x) .^{.^{.^{. \infty}}}}}

y = {(\sin x)}^{y}

\ln y = y \ln (\sin x)

\frac{1}{y} \cdot \frac{d y}{d x} = y \frac{\cos x}{\sin x} + (\ln \sin x) \frac{d y}{d x}

(\frac{1}{y} - \ln (\sin x)) \frac{d y}{d x} = \frac{y \cos x}{\sin x}

\frac{d y}{d x} = \frac{y^{2} \cos x}{\sin x} \times \frac{1}{1 - y \sin x}

Commented by rita1608 last updated on 11/Dec/17

t h a n k u s i r

Commented by mrW1 last updated on 11/Dec/17

A t r y t o s o l v e y :

y = {(\sin x)}^{y}

\Rightarrow y = e^{y \ln (\sin x)}

\Rightarrow y \times e^{- y \ln (\sin x)} = 1

\Rightarrow - y \ln (\sin x) \times e^{- y \ln (\sin x)} = - \ln (\sin x)

\Rightarrow - y \ln (\sin x) = W (- \ln (\sin x))

\Rightarrow y = - \frac{W (- \ln (\sin x))}{\ln (\sin x)}

i n g e n e r a l :

f {(x)}^{f {(x)}^{f {(x)}^{\dots}}} = - \frac{W (- \ln f (x))}{\ln f (x)}

f (x) > 0

\ln f (x) < \frac{1}{e}

\Rightarrow 0 < f (x) < e^{\frac{1}{e}} \approx 1.444

Answered by ibraheem160 last updated on 11/Dec/17