Question

If y = sin-1(x/a) prove that: dy/dx = 1/√a2 - x2 If y = cos-1(x/a) prove that: dy/dx = - 1/√a2 - x2 If y = tan-1(x/a) prove that: dy/dx = a/a2 + x2

Answer 1

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Answer 2

y = sin-1 x

Differentiate with respect to x,

dy / dx = 1 / √1 - x 2  (√1 - x 2 ) (dy / dx) = 1

Again differentiating with respect to x,

(√1 - x 2 ) (d2y / dx2 ) + (dy / dx) (1 / 2 √1 - x 2 ) * (d / dx) (1 - x 2 ) (√1 - x 2 ) (d2y / dx2 ) + (dy / dx) (-2x / 2√1 - x 2 ) = 0

(√1 - x 2 ) (d2y / dx2 ) = (dy / dx) (x / √1 - x 2 )

(√1 - x 2 ) (d2y / dx2 ) = (dy / dx) x

(1 - x 2 ) (d2y / dx2 ) = x (dy / dx)

(1 - x 2 ) (d2y / dx2 ) - x (dy / dx) = 0

$\rule{200}{2}$