Differentiate the following function with respect to `x :` `tan^(-1){(sqrt(1+a^2x^2)-1)/(a x)},x!=0`

Explanation

Let `y=tan ^{-1}(frac{sqrt{1+a^{2} x^{2}}-1}{a x})`

put `a x=tan theta`

`therefore y=tan ^{-1}(frac{sqrt{1+tan ^{2} theta}}{tan theta}. `

`Rightarrow y=tan ^{-1}(frac{sec theta-1}{tan theta}) `

`Rightarrow y=tan ^{-1}(frac{1-cos theta}{sin theta}) `

`Rightarrow y=tan ^{-1}(frac{2 sin ^{2} frac{theta}{2}}{2 frac{theta}{2} frac{theta}{2}})`

`Rightarrow y=tan ^{-1}(tan frac{theta}{2}) `

`Rightarrow y=frac{theta}{2} `

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