A function in which and are expressed as a function of a third variable is called a parametric function. For example, the function defined by the equations and is a parametric function.
Now we shall give an example to find the second derivative of the parametric function.
Example: If the parametric function , , then show that
We have the given parametric function
Differentiating both sides of equation (i) with respect to , we have
Differentiating both sides of equation (ii) with respect to , we have
Using the chain rule of differentiation , we have
Putting the values of and in the above chain rule formula, we have
Again, differentiating both sides with respect to , we have
Using the values of , we get
Putting the value of in the above result, we have