Question

(ii) If y = (tan-1 x)2, prove that (x2 + 1)2 + 2 x (x2 + 1)

Railway Exams

Maths

Solution

100

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( y=left(tan ^{-1} xright)^{2} ) ( D D r+x ) ( frac{d y}{d x}=2 tan ^{-1} x cdot frac{1}{1+x^{2}} ) ( frac{d y}{d x}=frac{2 tan ^{-1} x}{1+x^{2}}=0 ) ( frac{d y}{d x}left(1+x^{2}right)=2 tan ^{-1} x ) ( D D r t r ) ( frac{d^{2} y}{d x^{2}}left(1+x^{2}right)+frac{d y}{d x}(2 x)=2 cdot frac{1}{1+x^{2}} ) ( frac{d^{2} y}{d x^{2}}left(1+x^{2}right)+frac{2 tan ^{-1} x}{1+x^{2}} cdot(2 x)=frac{2}{1+x^{2}} ) ( frac{d^{2} y}{d x^{2}}left(1+x^{2}right)^{2}+4 x tan x=frac{2left(1+x^{2}right)}{left(x^{2}right)^{2}} ) ( left(1+x^{2}right)^{2} frac{d^{2} y}{d x^{2}}+2 xleft(1+x^{2}right) frac{d y}{d x}=2 ) ( f(n)left(2 tan ^{-1} x=left(1+x^{2}right) frac{d y}{d x}right. )