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

{y_2} = - \frac{{{a^2}}}{{{y^2}}}

We have the given parametric function

\begin{gathered} x = a\cos \theta \,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ y = a\sin \theta \,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right) \\ \end{gathered}

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

\begin{gathered} \frac{{dx}}{{d\theta }} = - a\sin \theta \\ \Rightarrow \frac{{d\theta }}{{dx}} = - \frac{1}{{a\sin \theta }} \\ \end{gathered}

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

\frac{{dy}}{{d\theta }} = a\cos \theta

Using the chain rule of differentiation , we have

\frac{{dy}}{{dx}} = \frac{{dy}}{{d\theta }} \times \frac{{d\theta }}{{dx}}

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

\frac{{dy}}{{dx}} = a\cos \theta \times \left( { - \frac{1}{{a\sin \theta }}} \right) = - \cot \theta

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

\begin{gathered} \frac{{{d^2}y}}{{d{x^2}}} = - \frac{d}{{dx}}\cot \theta \\ \Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = - {\csc ^2}\theta \frac{{d\theta }}{{dx}} \\ \end{gathered}

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

\begin{gathered} \frac{{{d^2}y}}{{d{x^2}}} = - {\csc ^2}\theta \left( { - \frac{1}{{a\sin \theta }}} \right) \\ \Rightarrow {y_2} = - \frac{1}{{{{\sin }^2}\theta }}\left( { - \frac{1}{{a\sin \theta }}} \right) \\ \Rightarrow {y_2} = - \frac{1}{{a{{\sin }^3}\theta }} = - \frac{{{a^2}}}{{{a^3}{{\sin }^3}\theta }} \\ \Rightarrow {y_2} = - \frac{{{a^2}}}{{{{\left( {a\sin \theta } \right)}^3}}} \\ \end{gathered}

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

{y_2} = - \frac{{{a^2}}}{{{y^3}}}