Question

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

Answer 1

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

Explanation:

For a parametric function `f(t)=(x(t),y(t))`

derivative `(dy)/(dx)` is defined as `(dy)/(dx)=((dy)/(dt))/((dx)/(dt))`

Here `x(t)=t/sint` hence `(dx)/(dt)=(sint-tcost)/sin^2t`

and as `y(t)=cost/t^2`, `(dy)/(dt)=(-t^2sint-2tcost)/t^4`

Hence `(dy)/(dx)=-((t^2sint+2tcost)/t^4)/((sint-tcost)/sin^2t)`

= `(sin^2t(t^2sint+2tcost))/(t^4(sint-tcost))`

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