# Second Derivative of the Parametric Equation

Parametric Function
A function in which $x$ and $y$ are expressed as a function of a third variable is called a parametric function. For example, the function defined by the equations $x = a{t^2}$ and $y = 2at$ is a parametric function.

Now we shall give an example to find the second derivative of the parametric function.

Example: If the parametric function $x = a\cos \theta$, $y = a\sin \theta$, then show that

We have the given parametric function

Differentiating both sides of equation (i) with respect to $\theta$, we have

Differentiating both sides of equation (ii) with respect to $\theta$, we have

Using the chain rule of differentiation , we have

Putting the values of $\frac{{d\theta }}{{dx}}$ and $\frac{{dy}}{{d\theta }}$ in the above chain rule formula, we have

Again, differentiating both sides with respect to $\theta$, we have

Using the values of $\frac{{d\theta }}{{dx}}$, we get

Putting the value of $a\sin \theta$ in the above result, we have