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Question
Differentiate the following w.r.t. x : `tan^-1((2^x)/(1 + 2^(2x + 1)))`
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Solution
Let y = `tan^-1((2^x)/(1 + 2^(2x + 1)))`
= `tan^-1[(2.2^x - 2^x)/(1 + (2.2^x)(2^x))]`
= tan–1(2.2x) – tan–1(2x)
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[tan^-1(2.2^x) - tan^-1(2^x)]`
= `"d"/"dx"[tan^-1(2.2^x)] - "d"/"dx"[tan^-1(2^x)]`
= `(1)/(1 + (2.2^x)^2)."d"/"dx"(2.2^x) - (1)/(1 + (2^x)^2)."d"/"dx"(2^x)`
= `(1)/(1 + 4(2^(2x))) xx 2 xx 2^xlog2 - (1)/(1 + 2^(2x)) xx 2^xlog2`
= `2^xlog2[2/(1 + 4(2^(2x))) - 1/(1 + 2^(2x))]`.
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