A function y = φ  (x) is said to be defined parametrically if both x and y are given in terms of a new variable, say t. The variable t is called the parameter.

Let x = f (t) and y = g (t) be two differentiable functions of ‘t’. We assume that x = f (t) possesses an inverse function t = h (x). Thus we obtain y = g (h (x)) and as such y can be considered as a composite function. Hence by the chain rule

dy/dx = dy/dt . dt/dx = dy/dt , provided dx/dt ≠  0.
                                 dx/dt

Example. Find dy/dx if

(i)    x =  1-t /1+t and y = 2t³ + 4^t.

(ii)    x = a log t and y = bt².

Solution. (i) We have

x = 1-t/1+t and  y = 2t³ + 4^t

dx = (1+t)(-1) – (1-t)(1)           and  dy/dt = 6t² + 4^t log 4
dt          (1+t)
i.e., dx/dt =  -2                           and  dy/dt = 6t² + 4^t log 4
                (1+t)

Thus

dy/dx = dy/dt  = (6t² + 4^t log 4) = (6t² + 4^t log 4) (1+t)²

(iii)    We have

x = a log t         and      y = bt²

dx/dt = a/t         and      dy/dt = 2bt

Thus      dy/dx = dy/dt   = 2 bt²/a .

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