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If, y = (sin^-1x)^2, then what is the value of (1 – x^2)y” – xy’ + 4?(a) 2(b) 4(c) 6(d) 8This question was posed to me during a job interview.Question is from Second Order Derivative topic in division Limits and Derivatives of Mathematics – Class 11

Answer»

Correct CHOICE is (c) 6

To explain: We have, y = (sin^-1x)^2 ………..(1)

Differentiating with respect to x, we get,

y’ = 2(sin^-1x)*1/(1 – x^2)^1/2

or, y’(1 – x^2)^1/2 = 2(sin^-1x)

Squaring both SIDES,

(1 – x^2)(y’)^2 = 4(sin^-1x)^2

From (1),

(1 – x^2)( y’)^2 = 4y

Differentiating with respect to x, we get

(1 – x^2)y” + (y’)^2 d/dx(1 – x^2) = 4 y’

=>(1 – x^2)2y’y” + (y’)^2 (-2X) = 4y’

Or, (1 – x^2)y” – XY’ = 2

Or, (1 – x^2)y” – xy’ + 4 = 2 + 4 = 6


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