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Question
Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`
Solution
Let y = `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`
= `tan^-1[(3sqrt(x) - xsqrt(x))/(1 - 3x)]`
Put `sqrt(x) = tanθ`. Thenθ = `tan^-1(sqrt(x))`
∴ y = `tan^-1((3tanθ - tan^3θ)/(1 - 3tan^2θ))`
= tan–1 (tan3θ)
= 3θ
= `3tan^-1(sqrt(x))`
∴ `"dy"/"dx" = 3"d"/"dx"[tan^-1(sqrt(x))]`
= `3 xx (1)/(1 + (sqrt(x))^2)."d"/"dx"(sqrt(x))`
= `(3)/(1 + x) xx (1)/(2sqrt(x))`
= `(3)/(2sqrt(x)(1 + x)`.
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