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Question
Differentiate the following w.r.t.x:
`cot^-1((1 + 35x^2)/(2x))`
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Solution
Let y = `cot^-1((1 + 35x^2)/(2x))`
= `tan^-1((2x)/(1 + 35x^2)) ...[∵ cot^-1 x = tan^-1(1/x)]`
= `tan^-1[(7x - 5x)/(1 + (7x)(5x))]`
= tan–1(7x) – tan–1(5x)
Differentiating w.r.t.x, we get
`(dy)/dx = d/dx [tan^-1 (7x) - tan^-1(5x)]`
= `d/dx [tan^-1(7x)] - d/dx [tan^-1(5x)]`
= `1/(1 + (7x)^2) * d/dx (7x) - 1/(1 + (5x)^2) * d/dx (5x)`
= `1/(1 + 49x^2) xx 7 - 1/(1 + 25x^2) xx 5`
= `7/(1 + 49x^2) - 5/(1 + 25x^2)`
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