Question

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`.

Answer 1

Now , given ` y = sin^(-1)x, 1 - le x le 1, |(x -pi)/(2)| le y le (pi)/(2)`
`therefore" "x = sin y " ".......(i)`
Differrentiate equation (1) w.r.t.y
`(dx)/(dy) = cos y`
`= pm sqrt(1- sin^(2)y)`
` = pm sqrt(1-x^(2))" "[by(1)]`
Since `(-pi)/(2) le y le (pi)/(2)`
`therefore ` ylies in I or IV quadrant
`therefore` cos y is positive .
`therefore " "(dx)/(dy) = sqrt(1-x^(2))`
we have , `(dy)/(dx) = (1)/(((dx)/(dy)))`
`therefore" "(dy)/(dx) = (1)/(sqrt(1-x^(2))), |x|lt 1` Hence Proved.