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Question
Differentiate the following:
y = `sin^-1 ((1 - x^2)/(1 + x^2))`
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Solution
y = `sin^-1 ((1 - x^2)/(1 + x^2))`
y = f(g(x)
`("d"y)/("d"x)` = f'(g(x)) . g'(x)
`("d"y)/("d"x) = 1/sqrt(1 - ((1 - x^2)/(1 + x^2))^2) xx "d"/("d"x) ((1 - x^2)/(1 + x^2))`
= `1/sqrt(((1 + x^2)^2 - (1 - x^2)^2)/(1 + x^2)^2) xx ((1 + x^2)(- 2x) - (1 - x^2)(2x))/(1 + x^2)^2`
= `1/sqrt((1 + 2x^2 + x^4 - (1 - 2x^2 + x^4))/((1 + x^2))) xx (-2x - 2x^3 - 2x + 2x^3)/(1 + x^2)^2`
= `(1 + x^2)/sqrt(1 + 2x^2 + x^4 - 1 + 2x^2 - x^4) xx (-4x)/(1 + x^2)^2`
= `(-4x)/(sqrt(4x^2) (1 + x^2))`
`("d"y)/("d"x) = (-4x)/(2x(1 + x^2))`
= `- 2/(1 + x^2)`
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