Question

How do you find `dy/dx` if `x=e^t/t` and `y=sqrte^t`?

Answer 1

`(dy)/(dx) =t^2/(2(t-1)sqrt(e^t))`

Explanation:

You can calculate the two differentials separately and then take the quotient:

`x =e^t/t => dx = d/(dt) (e^t/t)dt = (e^t/t-e^t/t^2)dt = e^t/t^2(t-1)dt`

`y=sqrt(e^t) => dy = d/(dt) (sqrt(e^t))dt = e^t/(2sqrt(e^t))dt = 1/2 sqrt(e^t)dt`

So:

`(dy)/(dx) = 1/2 (sqrt(e^t)dt)/(e^t/t^2(t-1)dt) = t^2/(2(t-1)sqrt(e^t))`