Question

How do you differentiate the following parametric equation: `x(t)=tlnt, y(t)= cost-tsin^2t`?

Answer 1

`(df(t))/dt = (ln(t) + 1 , -sin(t) - sin^2(t) - 2tsin(t)cos(t))`

Explanation:

Differentiating a parametric equation is as easy as differentiating each individual equation for its components.

If `f(t) = (x(t), y(t))` then `(df(t))/dt = ((dx(t))/dt , (dy(t))/dt)`

So we first determine our component derivatives:
`(dx(t))/dt = ln(t) + t/t = ln(t) + 1`
`(dy(t))/dt = -sin(t) - sin^2(t) - 2tsin(t)cos(t)`

Therefore the final parametric curve's derivatives is simply a vector of the derivatives:
`(df(t))/dt = ((dx(t))/dt , (dy(t))/dt)`
`= (ln(t) + 1 , -sin(t) - sin^2(t) - 2tsin(t)cos(t))`