Question

How do you differentiate the following parametric equation: `x(t)=3(t+1)^2+2e^t, y(t)= (t+2)^2+t^2`?

Answer 1

`(dy)/(dx) = (3(t+1))/(2t+1)`

Explanation:

Step 1: Find `(dx)/(dt)`
Step 2: Find `(dy)/(dt)`
3) Step 3: `(dy)/(dt) = (dx)/(dt) * (dy)/(dx) hArr (dy)/(dx)= [(dy)/(dt)]/[(dx)/(dt)]`

Step 1: Given `x(t)= 3(t+1)^2`
`x'(t) = (dx)/(dt)= 3(2)(t+1)(1)`

`color(red)((dx)/(dt)= 6(t+1))`

Step 2: Given `y(t)= (t+2)^2 +t^2`
`y'(t) = (dy)/(dt)= 2(t+2)(1)+2t`

`(dy)/(dt)= 2t+4+2t = 4t+2`
`color(blue)((dy)/(dt)= 4t+2 = 2(2t+1))`

Step 3: `(dy)/(dt) = (dx)/(dt) * (dy)/(dx) hArr (dy)/(dx)= [(dy)/(dt)]/[(dx)/(dt)]`

`(dy)/(dx)= color(red)(6(t+1))/color(blue)(2(2t+1))`

`(dy)/(dx) = (3(t+1))/(2t+1)`