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If x and y are differentiable functions of t, then `(dy)/(dx)=(dy//dt)/(dx//dt)," if "(dx)/(dt)ne0`.

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Let `deltax` and `deltay` be the increments in x and y respectively corresponding to the increment `deltat` in t.
Since x and y are differentiable functions of t,
`(dx)/(dt)=underset(deltatrarr0)lim(deltax)/(deltat) and (dy)/(dt)=underset(deltatrarr0)lim(deltay)/(deltat)" ...(1)"`
Also, as `deltatrarr0, deltaxrarr0" ...(2)"`
Now, `(deltay)/(deltax)=((deltay//deltat))/((deltax//deltat))" ..."[deltatne0]`
Taking limits as `deltatrarr0`, we get,
`underset(deltatrarr0)lim(deltay)/(deltax)=underset(deltatrarr0)lim((deltay//deltat))/((deltax//deltat))`
`underset(deltaxrarr0)lim(deltay)/(deltax)=(underset(deltatrarr0)lim(deltay//deltat))/(underset(deltatrarr0)lim(deltax//deltat))=((dy//dt))/((dx//dt))" ...[By (1) and (2)]"`
`because` the limits in R.H.S. exist
`therefore underset(deltaxrarr0)lim(deltay)/(deltax)` exists and is equal to `(dy)/(dx)`
`therefore (dy)/(dx)=(dy//dt)/(dx//dt)," if "(dx)/(dt)ne0`.
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