Question

How do you differentiate the following parametric equation: `x(t)=-t^2+tcost, y(t)=t^2sint`?

Answer 1

`dx/dt = cost -tsint -t`

`dy/dt =t^2cost + 2tsint`

`dy/dx = (t^2cost + 2tsint)/(cost -tsint - 2t)`

Explanation:

We have two parametric equations:

`x(t) = -t^2+tcost`
`y(t) = t^2sint`

We can differentiate both equation wrt `t` (and apply product rule);

`dx/dt = -2t+(t)(-sint)+(1)(cost)`
`\ \ \ \ \ = cost -tsint - 2t`

`dy/dt = (t^2)(cost) + (2t)(sint)`
`\ \ \ \ \ = t^2cost + 2tsint`

And presumably you also want `dy/dx` which we get from the chain rule:

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

`\ \ \ \ \ = (t^2cost + 2tsint)/(cost -tsint - 2t)`