Question

What is the derivative of `f(t) = (t/sint , t^2-t^3cost )`?

Answer 1

`(df)/(dx)=(2tsin^2t-3t^2sin^2tcost+t^3sin^3t)/(sint-tcost)`

Explanation:

In parametruic form of function such as `f(t)=(x(t),y(t))`

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

Now as `y(t)=t^2-t^3cost`, `(dy)/(dt)=2t-3t^2cost+t^3sint`

and as `x(t)=t/sint`, `(dx)/(dt)=(sint-tcost)/(sin^2t)`

Hence `(df)/(dx)=(2t-3t^2cost+t^3sint)/((sint-tcost)/(sin^2t))`

= `(2tsin^2t-3t^2sin^2tcost+t^3sin^3t)/(sint-tcost)`