Question

How do you find d/dt given `y=t^3 - 3t` and `x=t^2`?

Answer 1

It depends upon what you intended by `d/(dt)`

See possible interpretations below:

Explanation:

Option 1: You wanted `color(black)((dcolor(red)y)/(dt))`
`(dy)/(dt)=3t^2-3` (using the exponent rule)

Option 2: You wanted `color(black)((dcolor(red)x)/(dt))`
`(dx)/(dt)=2t` (using the exponent rule)

Option 3: You wanted `color(black)((dy)/(dcolor(red)x)`
...definitely not what was asked for, but a possibly intended requirement.
If `y=t^3-3t` and `x=t^2`
then `t=x^(1/2)` and `t^3=x^(3/2)`
so
`y=x^(3/2)-3x^(1/2)`
and
`(dy)/(dx)= 3/2x^(1/2) -3/2x^(-1/2) = (3sqrt(x))/2-3/(2sqrt(x))`