Differentiate the following function with respect to `x :` `tan^(-1)((a+btanx)/(b-atanx))`

Explanation

Solution:

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

`\Rightarrow y=\tan ^{-1}(\frac{\frac{a}{b}+\tan x}{1-\frac{a}{b} \tan x}) \ldots`
[Dividing numerator and denominator by b ] Now, `\tan ^{-1}(\frac{m+n}{1-m n})=\tan ^{-1} m+\tan ^{-1} n`

`\therefore y=\tan ^{-1}(\frac{a}{b}\right)+\tan ^{-1}(\tan x)`

`\Rightarrow y=\tan ^{-1}(\frac{a}{b}\right)+x`

Now, Differentiating both sides w.r.t. x,

`\Rightarrow \frac{d y}{d x}=0+1 `

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