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Question
Differentiate the following w.r.t. x:
`tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
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Solution
Let y = `tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
= `tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x)) ...[∵ cot^-1x = tan^-1(1/x)]`
= `tan^-1(x/(1 + 6x^2)) + tan^-1((7x)/(1 - 10x^2)) ...[∵ cot^-1x = tan^-1(1/x)]`
= `tan^-1[(3x - 2x)/(1 + (3)(2x))] + tan^-1[(5x + 2x)/(1 - (5x)(2x))]`
= tan–13x – tan–12x + tan–15x + tan–12x
= tan–13x + tan–15x
∴ `"dy"/"dx" = "d"/"dx"[tan^-1 3x + tan^-1 5x]`
= `"d"/"dx"(tan^-1 3x) + "d"/"dx"(tan^-1 5x)`
= `(1)/(1 + (3x)^2)."d"/"dx"(3x) + (1)/(1 + (5x)^2)."d"/"dx"(5x)`
= `(1)/(1 + 9x^2) xx 3 xx 1 + (1)/(1 + 25x^2) xx 5 xx 1`
= `(3)/(1 + 9x^2) + (5)/(1 + 25x^2`
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