Question

What is the derivative of `f(t) = (t +e^t, e^t-tcost )`?

Answer 1

`(dy)/(dx)=(e^t+tsint)/(1+e^t)`

Explanation:

`x=t+e^t`
`y=e^t-tcost`

`(dy)/(dx)=(dy)/(dt)-:(dx)/(dt)`

`(dy)/(dt)=d/dt[e^t-tcost]`
`color(white)((dy)/(dt))=d/dt[e^t]-d/dt[tcost]`
`color(white)((dy)/(dt))=e^t+tsint`

`(dx)/(dt)=d/dt[t+e^t]`
`color(white)((dx)/(dt))=d/dt[t]+d/dt[e^t]`
`color(white)((dx)/(dt))=1+e^t`

`(dy)/(dx)=(e^t+tsint)/(1+e^t)`