Question

Find $ \dfrac{{dy}}{{dx}} $ ,
If $ \dfrac{{dx}}{{dt}} = ap\cos pt,\,\,\dfrac{{dy}}{{dt}} = - bp\sin pt $

Answer 1

Hint: Here, derivative of two parametric equations are given. To find a solution to the required problem or required value of $ \dfrac{{dy}}{{dx}} $ from them we divide derivatives of parametric equations having y variables with derivatives of parametric equations having x variables.

Complete step-by-step answer:
Given, derivative of two parametric equations. Which are as follows:
 $ \dfrac{{dx}}{{dt}} = ap\cos pt\,\,and\,\,\,\dfrac{{dy}}{{dt}} = - bp\sin pt $
To find $ \dfrac{{dy}}{{dx}} $ . We divide two parametric equations. As, we required $ \dfrac{{dy}}{{dx}} $ as an answer. We divide derivatives of parametric equations having y variables with derivatives of parametric equations having x variables.
Which implies we have
\[
  \dfrac{{\dfrac{{dy}}{{dt}}}}{{\dfrac{{dx}}{{dt}}}} = \dfrac{{ - bp\sin pt}}{{ap\cos pt}} \\
   \Rightarrow \dfrac{{dy}}{{dt}} \times \dfrac{{dt}}{{dx}} = \dfrac{{ - b\sin pt}}{{a\cos pt}} \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{ - b\sin pt}}{{a\cos pt}} \\
   \Rightarrow \dfrac{{dy}}{{dx}} = - \dfrac{b}{a}\tan pt \;
\]
Hence, from above we see that the value of $ \dfrac{{dy}}{{dx}} $ is $ \dfrac{{ - b}}{a}\tan pt $ .

Note: To find the value of $ \dfrac{{dy}}{{dx}} $ . We first see that what function is given in the problem if there is a function y given in terms of ‘x’ then we can find the value of $ \dfrac{{dy}}{{dx}} $ directly. But, in case when derivative of two parametric equations are given then on dividing derivative of parametric equation having y variable with another equation we can find the value of $ \dfrac{{dy}}{{dx}} $ .