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Question
Differentiate the following w.r.t. x:
sin (tan–1 e–x)
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Solution
Let, y = sin (tan–1 e–x)
Differentiating both sides with respect to x,
`dy/dx = d/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/(1 + e^(-2x)) (e^-x) d/dx (-x)`
= `cos (tan^-1 e^-x) e^-x/(1 + e^(-2x)) (-1)`
= `-(e^-x cos (tan^-1 e^-x))/(1 + e^(-2x))`
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