# Second Derivative of Parametric Equation

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

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

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

We have 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 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

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

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