Differentiate the following with respect of x: y = \tan^{-1}(\sqrt{1+x}-\sqrt{1-x}/\sqrt{1+x}+\sqrt{1-x})

CBSE Class 12 Math 2008 Solved Paper - Question 15

Question : 15 of 29

Marks: +1, -0

Differentiate the following with respect of x:
y =

tan^{-1}({√{1+x}-√{1-x}}/{√{1+x}+√{1-x}})

Solution:

Let x = cos 2θ ⇒ θ =

1/2 cos^{-1}

x ... 1
∴

√{1+x}

√{1+cos2θ}

√{1+2cos^2θ-1}

√2

cos θ

√{1-x}

√{1-cos2θ}

√{1-1-2sin^2θ}

√2

sin θ
Let y =

tan^{-1}|{√{1+x}-√{1-x}}/{√{1+x}+√{1-x}}|

tan^{-1}|{√2cosθ-√2sinθ}/{√2cosθ+√2sinθ}|

tan^{-1}|{1-tanθ}/{1+tanθ}|

tan^{-1} \{tan(π/4-θ)\}

π/4

- θ =

π/4

1/2 cos^{-1}

x From 1
∴

{dy}/{dx}

-1/2(-1/√{1-x^2})

1/{2√{1-x^2}}

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