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Question
Differentiate the following w.r.t. x:
`tan^-1((2x^(5/2))/(1 - x^5))`
Solution
Let y = `tan^-1((2x^(5/2))/(1 - x^5))`
Put `x^(5/2)` = tanθ
Then θ = `tan^-1(x^(5/2))`
∴ y = `tan^-1((2tanθ)/(1 - tan^2θ))`
= tan–1(tan2θ)
= 2θ
= `2tan^(–1)(x^(5/2))`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[2tan^-1(x^(5/2))]`
= `2"d"/"dx"[tan^-1(x^(5/2))]`
= `2 xx (1)/(1 + (x^(5/2))^2)."d"/"dx"(x^(5/2))`
= `(2)/(1+ x^5) xx (5)/(2)x^(3/2)`
= `(5xsqrt(x))/(1 + x^5)`
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