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Question
Differentiate the following w.r.t. x : `tan^-1((2sqrt(x))/(1 + 3x))`
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Solution
Let y = `tan^-1((2sqrt(x))/(1 + 3x))`
= `tan^-1[(3sqrt(x) - sqrt(x))/(1 + (3sqrt(x))(sqrt(x))}]`
= `tan^-1(3sqrt(x)) - tan^-1(sqrt(x))`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[tan^-1(sqrt(x)) - tan^-1(sqrt(x))]]`
= `"d"/"dx"[tan^-(3sqrt(x)] - "d"/"dx"[tan^-1(sqrt(x))]`
= `(1)/(1 + (3sqrt(x))^2)."d"/"dx"(3sqrt(x)) - (1)/(1 + (sqrt(x))^2)."d"/"dx"(sqrt(x))`
= `(1)/(1 + 9x) xx 3 xx (1)/(2sqrt(x)) - (1)/(1 + x) xx (1)/(2sqrt(x)`
= `(1)/(2sqrt(x))[3/(1 + 9x) - 1/(1 + x)]`.
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