Question

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

Answer 1

`dx/dt = te^tcost + te^tsint + e^tsint`

`dy/dt = -tsint + cost-2sintcost`

Explanation:

We have parametric equations:

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

So, using the product rule and differentiating wrt `t` we have:

`x'(t) = (t)(e^t)(d/dt sint) + (t)(d/dt e^t)(sint) + (d/dt t)(e^t)(sint)`

`\ \ \ \ \ \ \ = (te^t)(cost) + (t)(e^t)(sint) + (1)(e^tsint)`

`\ \ \ \ \ \ \ = te^tcost + te^tsint + e^tsint`

And:

`y'(t) = (t)(d/dt cost) + (d/dt t)(cost)-d/dt sin^2t`

`\ \ \ \ \ \ \ = (t)(-sint) + (1)(cost)-2sint(cost)`

`\ \ \ \ \ \ \ = -tsint + cost-2sintcost`

This technically is the solution, but more likely we seek the full derivative:

`dy/dx = (dy//dt)/(dx//dt)`
`\ \ \ \ \ \ = (cost-tsint -2sintcost)/(te^tcost + te^tsint + e^tsint)`