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Find derivative of sin(tan^(-1)e^(-x)) w...

Find derivative of `sin(tan^(-1)e^(-x))` w.r.t. to `x`

Text Solution

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Let y =` sin (tan ^(-1) e^(-x))`
`rArr (dy)/(dx) sin (tan^(-1) e ^(-x))`
` = cos ( tan ^(-1)e^(-x)) (d)/(dx) (tan^(-1)e^(-x))`
`= cos (tan ^(-1)e^(-x)). (1)/(1+ (e^(-x))^(2))(d)/(dx) e^(-x)`
`= cos (tan ^(-1)e^(-x))/(1 + e^(-2x)).e ^(-x). (d)/(dx) (-x)`
`=-(e ^(x).cos (tan^(1) e^(x)))/(1+e^-2x) `
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