If $y=\big(\tan^{-1}x\big)^{2}$, show that $(x^{2}+1)^{2}\dfrac{d^{2}y}{dx^{2}}+2x(x^{2}+1)\dfrac{dy}{dx}=2$.

CBSE Class 12 · 2012 · DifferentiationLong Answer

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

, show that

(x^{2} + 1)^{2} \frac{d ^{2} y}{d x ^{2}} + 2 x (x^{2} + 1) \frac{d y}{d x} = 2

Answer: Proved

Differentiate once:

\frac{d y}{d x} = 2 tan^{- 1} x \cdot \frac{1}{1 + x ^{2}} \Rightarrow (1 + x^{2}) \frac{d y}{d x} = 2 tan^{- 1} x

Differentiate this relation again (product rule on the left):

(1 + x^{2}) \frac{d ^{2} y}{d x ^{2}} + 2 x \frac{d y}{d x} = \frac{2}{1 + x ^{2}}

Multiply throughout by

(1 + x^{2})

(x^{2} + 1)^{2} \frac{d ^{2} y}{d x ^{2}} + 2 x (x^{2} + 1) \frac{d y}{d x} = 2

CBSE Class 12 Board · 2012 · Subjective