If `y= cos^(-1) x`, Find `(d^(2)y)/(dx^(2))` in terms of y alone.

Explanation

`y=cos^(-1)x,`
`"or "x=cos y`
Differentiating w.r.t. y, we get
`(dx)/(dy)=-sin y`
`"or "(dy)/(dx)=-cosec y`
Differentiating w.r.t. x, we get
`(d^(2)y)/(dx^(2))=(d)/(dx)(-cosec y)`
`=(d)/(dy)=(-"cosec "y)(dy)/(dx)`
`="cosec y cot y (-cosec y)"`
`-cot ycdot cosec^(2) y`