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प्रश्न
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f-1(y) in the following: y = `sqrt(2 - sqrt(x)`
उत्तर
y = `sqrt(2 - sqrt(x)` ...(1)
We have to find the inverse function of y = f(x), i.e x in terms of y.
From (1),
y2 = `2 - sqrt(x) ∴ sqrt(x) = 2 - y^2`
∴ x = (2 – y2)2
∴ x = f-1(y) = (2 – y2)2
∴ `"dx"/"dy" = "d"/"dy"(2 - y^2)^2`
= `2(2 - y^2)."d"/"dy"(2 - y^2)`
= 2(2 – y2).(0 – 2y)
= –4y(2 – y2)
= `-4sqrt(2 - sqrt(x))(2 - 2 + sqrt(x))` ...[By (1)]
= `-4sqrt(x)sqrt(2 - sqrt(x)`
∴ `"dy"/"dx" = (1)/(("dx"/"dy")`
= `-(1)/(4sqrt(x)sqrt(2 - sqrt(x)`.
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