In our book of co-ordinate geometry, we have already learnt that the Cartesian equation of the conics can also be given in the parametric form. For example the parabola y2 = 4ax; the parametric from is given by y = 2at and x = at2. Therefore having established that a function 'y' of 'x' can be represented by the parametric equations as well. Lets say x = g(t) and y = h(t) are the parametric equations of y = f(x).
Now, let us assume that these functions are differentiable and the inverse of the functions x = g(t) is given by t = G(x).
f(x) = y = h(t) and t = G(x)
∴ f(x) = y = h(G(x))
Differentiating w.r.t x; we get
Also, Since x = g(t) and t = G(x) are the inverse functions of each other, Therefore
Substituting this ...