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Here is how to do it.
Works with the following version of Sage.
sage: version()
'SageMath version 9.0, Release Date: 2020-01-01'
Declare variable t and function v,
and define function F in terms of those:
sage: t = SR.var('t')
sage: v = function('v')
sage: F(t) = v(t)*t^2
Derivative of the function:
sage: F.diff(t)
t |--> t^2*diff(v(t), t) + 2*t*v(t)
Derivative of the expression:
sage: F(t).diff(t)
t^2*diff(v(t), t) + 2*t*v(t)
