Sometimes the relation between two variables is neither explicit nor implicit, but some link of a third variable with each of the two variables, separately, establishes a relation between the first two variables. In such a situation, we say that the relation between them is expressed via a third variable. The third variable is called the parameter. More precisely, a relation expressed between two variables x and y in the form x = f(t), y = g(t) is said to be parametric form with t as a parameter.
In order to find derivative of function in such form, we have by chain rule.
`(dy)/(dt) = (dy)/(dx) . (dx)/(dt)`
or `(dy)/(dx)` = `(((dy)/(dt))/((dx)/(dt)))` `("whenever" (dx)/(dt) ≠ 0)`
`(dy)/(dx) = (g'(t))/(f'(t)) (as (dy)/(dt) = g'(t) and (dx)/(dt) = f'(t))` [provided f'(t) ≠ 0]
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If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and hence, find dy/dx if x=a cost, y=a sint