Derivatives Of Functions In Parametric Forms

We have,

%5Cfrac%7Bx%5E%7B2%7D%7D%7Ba%5E%7B2%7D%7D%2B%5Cfrac%7By%5E%7B2%7D%7D%7Bb%5E%7B2%7D%7D%3D1

Let

x%3Da%20%5Ccos%20%5Ctheta%2C%20y%3Db%20%5Csin%20%5Ctheta

%5Ctherefore%20%5Cfrac%7Bd%20x%7D%7Bd%20%5Ctheta%7D%3D-a%20%5Csin%20%5Ctheta%2C%20%5Cfrac%7Bd%20y%7D%7Bd%20%5Ctheta%7D%3Db%20%5Ccos%20%5Ctheta

%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%3D-%5Cfrac%7Bb%7D%7Ba%7D%20%5Ccot%20%5Ctheta

On differentiating w.r.t.

%5Ctheta

, we get

%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D%3D-%5Cfrac%7Bb%7D%7Ba%7D%5Cleft(-%5Coperatorname%7Bcosec%7D%5E%7B2%7D%20%5Ctheta%5Cright)%20%5Cfrac%7Bd%20%5Ctheta%7D%7Bd%20x%7D

%5CRightarrow%20%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D%3D%5Cfrac%7Bb%20%5Coperatorname%7Bcosec%7D%5E%7B2%7D%20%5Ctheta%7D%7B-a%5E%7B2%7D%20%5Csin%20%5Ctheta%7D

%5CRightarrow%20%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D%3D-%5Cfrac%7Bb%7D%7Ba%5E%7B2%7D%20%5Csin%20%5E%7B3%7D%20%5Ctheta%7D

%5CRightarrow%20%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%7D%3D%5Cfrac%7B-b%5E%7B4%7D%7D%7Ba%5E%7B2%7D%20y%5E%7B3%7D%7D%20%5B%5Cbecause%20%5Csin%20%5Ctheta%3D%5Cfrac%7By%7D%7Bb%7D%5D

Chapter Questions
1 MCQs