Question

How do you differentiate the following parametric equation: `x(t)=tlnt, y(t)= t^3cost-tsin^2t`?

Answer 1

`dy/dx = (-t^3sint + 3t^2cost - tsin2t -sin^2t) /(1 + lnt)`

Explanation:

We have two parametric equations:

`x = tlnt`
`y = t^3cost - tsin^2t`

We differentiate wrt `t` (and apply the product rule):

`dx/dt = (t)(1/t) + (1)(lnt)`
`\ \ \ \ \ \ = 1 + lnt`

And:

`dy/dt = t^3cost - tsin^2t`
`\ \ \ \ \ \ = (t^3)(-sint) + (3t^2)(cost) - {(t)(2sintcost) + (1)(sin^2t)}`
`\ \ \ \ \ \ = -t^3sint + 3t^2cost - tsin2t -sin^2t`

And from the chain rule we have:

`dy/dx = (dy/dt)/(dy/dx)`
`\ \ \ \ \ \ = (-t^3sint + 3t^2cost - tsin2t -sin^2t) /(1 + lnt)`