InterviewSolution
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InterviewSolution
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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`. |
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Answer» 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. |
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