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If y = f(x) is a derivable function of x...

If y = f(x) is a derivable function of x such that the inverse function `x = f^(-1)(y)` is defined, then show that `(dx)/(dy)=(1)/((dy//dx))`, where `(dy)/(dx) ne 0`.

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Let `deltay` be the increment in y corresponding to an increment `deltax` in x. `therefore` as `deltaxrarr0, deltayrarr0`.
Now y is a differentiable function of x. `therefore underset(deltaxrarr0)lim(deltay)/(deltax)=(dy)/(dx)`
Now `(deltay)/(deltax)xx(deltax)/(deltay)=1" "therefore (deltax)/(deltay)=(1)/(((deltay)/(deltax)))`
Taking limits on both sides as `deltaxrarr0`, we get,
`underset(deltaxrarr0)lim(deltax)/(deltay)=underset(deltaxrarr0)lim[(1)/(((deltay)/(deltax)))]=(1)/(underset(deltaxrarr0)lim(deltay)/(deltax))`
`therefore underset(deltayrarr0)lim(deltax)/(deltay)=(1)/(underset(deltaxrarr0)lim(deltay)/(deltax))" ... "["as "deltaxrarr0, deltayrarr0]`
Since limit in R.H.S. exists
`therefore` limit in L.H.S. also exists and we have, `underset(deltayrarr0)lim(deltax)/(deltay)=(dx)/(dy)`
`therefore (dx)/(dy)=(1)/((dy//dx))`, where `(dy)/(dx) ne 0`.
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