Question

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

Answer 1

First, differentiate each individual function as you have been all year.

using the chain rule,

`x(t)=(t-2)^(1/2)`

`" "" "=>dx/dt=1/2(t-2)^(-1/2)*d/dt(t-2)`

`" "" "color(white)(=>dx/dt)=1/2(t-2)^(-1/2)*1`

`" "" "color(white)(=>dx/dt)=color(red)(1/(2sqrt(t-2))`

and, by the product rule,

`y(t)=t^2-2t^3e^t`

`" "" "=>dy/dt=2t-[(d/dt2t^3)e^t-2t^3(d/dte^t)]`

`" "" "color(white)(=>dy/dt)=2t-[6t^2e^t-2t^3e^t]`

`" "" "color(white)(=>dy/dt)=color(red)(2t(1-3te^t+t^2e^t)`

if we want to know `dy/dx`, we see that:

`dy/dx=(dy/dt)/(dx/dt)=(2t(1-3te^t+t^2e^t))/(1/(2sqrt(t-2)))=color(blue)(4tsqrt(t-2)(1-3te^t+t^2e^t)`