Differentiate the following function with respect to `x :` `t a n^(-1)((2^x+1)/(1-4^x)),oo

Explanation

Solution: `tan ^{-1}(frac{2^{x+1}}{1-4^{x}})`

Further simplifying we have,

Let `y=tan ^{-1}[frac{2 cdot 2^{x}}{1-(2^{x})^{2}}] quad` considering `2^{x}=tan theta` we get, `theta=tan ^{-1}(2^{x})`

Now,

``
`y=tan ^{-1}[frac{2 tan theta}{1-tan ^{2} theta}] `


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


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


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