If `y=logsqrt(tanx)` , write `(dy)/(dx)` .

Explanation

`text { We have, } {y}=log sqrt{tan x} `

`Rightarrow y=log (tan x)^{frac{1}{2}} `

`Rightarrow y=frac{1}{2} log tan x[because log a^{b}=b log a] `

`Rightarrow frac{d y}{d x}=frac{1}{2} times frac{1}{tan x} frac{d}{d x}(tan x) `

`Rightarrow frac{d y}{d x}=frac{1}{2} times frac{1}{tan x}(sec ^{2} x) `

`Rightarrow frac{d y}{d x}=frac{1}{2 frac{sin x}{cos x} times cos ^{2} x} `

`Rightarrow frac{d y}{d x}=frac{1}{2 sin x cos x} `

`Rightarrow frac{d y}{d x}=frac{1}{sin 2 x} `

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