# Derivatives of Functions in Parametric Forms

#### notes

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)

Thus

(dy)/(dx) =   (g'(t))/(f'(t))  (as (dy)/(dt) = g'(t) and (dx)/(dt) = f'(t)) [provided f'(t) ≠  0]

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