Question

If y=cos^-1 x find d^2y/dx^2 in terms of y alone.

Answer 1

this will help u dear

Answer 2

Given, $\bf{y=cos^{-1}x}$
now differentiate y with respect to x,
dy/dx = -1/√(1 - x²) = -(1 - x²)^{-1/2}
differentiate once again,
d²y/dx² = 1/2(1 - x²)^{-3/2} (-2x)
= -x/√{(1 - x²)³} ------(1)

now, y = cos^{-1}x
cosy = x , use it in equation (1),

d²y/dx² = -cosy/√{(1 - cos²y)³}
= cosy/√(sin²y)³
= cosy/sin³y
= {cosy/siny}.{1/sin²y}
= coty.cosec²y

hence, d²y/dx² = coty.cosec²y