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

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


We have 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 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}


Putting 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}}}

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