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Question Number 54790 by Anju last updated on 11/Feb/19

d i f f e r e n c i a t e t h e f o l l o w i n g

i) x^{x} + {(s i n x)}^{l n x} =

i i) {s i n}^{- 1} (t a n h x) =

i i i) \sqrt{1 + x^{2} / 1 - x^{2} =}

Commented by maxmathsup by imad last updated on 11/Feb/19

1) l e t f (x) = x^{x} + {(s i n x)}^{l n (x)} \Rightarrow f (x) = e^{x l n (x)} + e^{l n (x) l n (s i n x)} w i t h x > 0 \Rightarrow

f^{^{'}} (x) = {(x l n x)}^{^{'}} x^{x} + {(l n x l n (s i n x))}^{^{'}} {(s i n x)}^{l n (x)}

= (l n x + 1) x^{x} + (\frac{l n (s i n x)}{x} + l n x \frac{c o s x}{s i n x}) {(s i n x)}^{l n (x)}

= (1 + l n (x)) x^{x} + (\frac{l n (s i n x)}{x} + l n x c o t a n (x)) {(s i n x)}^{l n (x)} .

Commented by maxmathsup by imad last updated on 11/Feb/19

i i) l e t g (x) = a r c s i n (t a n h x) w e h a v e t a n h x = \frac{s h (x)}{c h (x)} = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} = \frac{e^{2 x} - 1}{e^{2 x} + 1} \Rightarrow

g (x) = a r c s i n (\frac{e^{2 x} - 1}{e^{2 x} + 1}) \Rightarrow g^{^{'}} (x) = \frac{{(\frac{e^{2 x} - 1}{e^{2 x} + 1})}^{^{'}}}{\sqrt{1 - {(\frac{e^{2 x} - 1}{e^{2 x} + 1})}^{2}}} b u t

{(\frac{e^{2 x} - 1}{e^{2 x} + 1})}^{^{'}} = \frac{2 e^{2 x} (e^{2 x} + 1) - 2 e^{2 x} (e^{2 x} - 1)}{{(e^{2 x} + 1)}^{2}} = \frac{4 e^{2 x}}{{(e^{2 x} + 1)}^{2}} \Rightarrow

g^{^{'}} (x) = \frac{4 e^{2 x}}{{(e^{2 x} + 1)}^{2} \frac{\sqrt{{(e^{2 x} + 1)}^{2} - {(e^{2 x} - 1)}^{2}}}{e^{2 x} + 1}} = \frac{4 e^{2 x}}{(e^{2 x} + 1) \sqrt{e^{4 x} + 2 e^{2 x} + 1 - e^{4 x} + 2 e^{2 x} - 1}}

= \frac{4 e^{2 x}}{(1 + e^{2 x}) 2 e^{x}} = \frac{2 e^{x}}{(1 + e^{2 x})} \Rightarrow g^{^{'}} (x) = \frac{2 e^{x}}{(1 + e^{2 x})} .

Answered by tanmay.chaudhury50@gmail.com last updated on 11/Feb/19

1) y = x^{x} + {(s i n x)}^{l n x}

y = u + v

\frac{d y}{d x} = \frac{d u}{d x} + \frac{d v}{d x}

u = x^{x}

l n u = x l n x

\frac{1}{u} \frac{d u}{d x} = x \times \frac{1}{x} + l n x \times 1

\frac{d u}{d x} = x^{x} (1 + l n x)

v = {(s i n x)}^{l n x}

l n v = l n x l n (s i n x)

\frac{1}{v} \times \frac{d v}{d x} = l n x \times \frac{1}{s i n x} \times c o s x + l n (s i n x) \times \frac{1}{x}

\frac{d v}{d x} = {(s i n x)}^{l n x} [c o t x l n x + \frac{l n (s i n x)}{x}]

s o a n s w e r i s

\frac{d u}{d x} + \frac{d v}{d x}

= x^{x} (1 + l n x) + {(s i n x)}^{l n x} [c o t x l n x + \frac{l n (s i n x)}{x}]

Answered by tanmay.chaudhury50@gmail.com last updated on 11/Feb/19

i i i) y = \sqrt{\frac{1 + x^{2}}{1 - x^{2}}}

l n y = \frac{1}{2} {l n (1 + x^{2}) - l n (1 - x^{2})}

\frac{1}{y} \frac{d y}{d x} = \frac{1}{2} {\frac{2 x}{1 + x^{2}} - \frac{- 2 x}{1 - x^{2}}}

\frac{1}{y} \frac{d y}{d x} = x {\frac{1 - x^{2} + 1 + x^{2}}{1 - x^{4}}}

\frac{d y}{d x} = \sqrt{\frac{1 + x^{2}}{1 - x^{2}}} (\frac{2 x}{1 - x^{4}})

Answered by tanmay.chaudhury50@gmail.com last updated on 11/Feb/19

i i) y = {s i n}^{- 1} (t a n h x)

\frac{d y}{d x} = \frac{1}{\sqrt{1 - {(t a n h x)}^{2}}} \times {s e c h}^{2} x