Question

How do you differentiate the following parametric equation: `x(t)=t^2+tcos2t, y(t)=tsint`?

Answer 1

`x'(t) = 2t - 4sin(2t)`

`y'(t) = sint + tcost`

Explanation:

We're asked to find the derivative of a set of parametric equations.

We differentiate each individual equation as normal:

x(t)

We can differentiate term by term, and factor out the constant,`2` in the second term:

`x'(t) = d/(dt) [x^2] + 2d/(dt) [cos(2t)]`

Use the power rule:

`d/(dt) [t^n] = nt^(n-1)`

where `n = 2`:

`= color(red)(2t) + 2d/(dt) [cos(2t)]`

Using the chain rule:

`d/(dt) [cos(2t)] = (dcosu)/(du)(du)/(dt)`

where `u = 2t` and `d/(du) [cosu] = -sinu`:

`= 2t + 2d/(dt)[2t]*-sin(2t)`

Simplifying:

`= 2t - 2d/(dt)[2t]sin(2t)`

Factor out the constant, `2`:

`= 2t - 4d/(dt)[t]sin(2t)`

The derivative of `x` is `1`

`color(red)(= 2t - 4sin(2t)`

y(t)

Use the product rule:

`d/(dt) [uv] = v(du)/(dt) + u(dv)/(dt)`

where `u = t` and `v = sint`:

`y'(t) = td/(dt) [sint] + d/(dt) [t] sint`

The derivative of `sint` is `cost`:

`= d/(dt)[t] sint +t cost`

The derivative of `t` is `1`:

`color(blue)(= sint + tcost`