If y = f(x) and x = g(y), where g is the inverse of f, i.e., `g = f^(-1)` and if `(dy)/(dx) and (dx)/(dy)` both exist and `(dx)/(dy) ne 0`, show that `(dy)/(dx) = (1)/((dx//dy))`.
Hence, (1) find `(d)/(dx) (tan^(-1)x)`
(2) If `y=sin^(-1)x, -1lexle1, -(pi)/(2)leyle(pi)/(2)`, then show that `(dy)/(dx)=(1)/(sqrt(1-x^(2)))` where `|x| lt 1`.

Explanation

Let `deltax` and `deltay` be the corresponding increments in x and y respectively.
`therefore " as "deltaxrarr0, deltayrarr0 and "as "deltayrarr0, deltaxrarr0`
Now, `(deltay)/(deltax)xx(deltax)/(deltay)=1." ... "[because deltax ne0, deltay ne0]`
Taking limits as `deltax rarr 0`, we get,
`underset(deltaxrarr0)lim((deltay)/(deltax)xx(deltax)/(deltay))=1" "therefore underset(deltaxrarr0)lim(deltay)/(deltax)xxunderset(deltaxrarr0)lim(deltax)/(deltay)=1`
`therefore underset(deltaxrarr0)lim(deltay)/(deltax)xxunderset(deltayrarr0)lim(deltax)/(deltay)=1" ... "["as "deltaxrarr0, deltayrarr0]`
`therefore (dy)/(dx).(dx)/(dy)=1" "...[because(dy)/(dx) and (dx)/(dy)" both exist"]`
`therefore (dy)/(dx)=(1)/((dx//dy)),"as "(dx)/(dy)ne0`.