Question

How do you differentiate the following parametric equation: `(t-t^3,3/t^4-t^3)`?

Answer 1

`(dy)/(dx)=(3t^7+12)/(3t^7-t^5)`

Explanation:

In such a parametric equation, where `(x,y)` are given by `(t-t^3,3/t^4-t^3)`

`(dy)/dx=(dy/dt)/(dx/dt)`

Now `(dy)/(dt)=d/dt(3/t^4-t^3)=d/dt(3t^(-4)-t^3)`

= `-12t^(-5)-3t^2=-12/t^5-3t^2`

and `(dx)/(dt)=d/dt(t-t^3)=1-3t^2`

Hence, `(dy)/(dx)=(-12/t^5-3t^2)/(1-3t^2)=(-12-3t^7)/(t^5-3t^7)=(3t^7+12)/(3t^7-t^5)`