The question
From practical experience, I know that the unit of an integral - resulting from integration of an expression with respect to a variable with a unit (i.e. non-dimensionless variable) - is the same as if the expression and the variable were multiplied.
Similarly, the unit of a derivative - resulting from differentiation of an expression with respect to a variable with a unit (i.e. non-dimensionless variable) - is the same as if the expression was divided by the variable.
This makes sense to me, and I have never really thought about it before. The question is: can this fact be regarded as a general (mathematical) rule? If yes; is it possible to give a compelling, rigorous and yet simple argument for this fact? If not; are there any good examples to illustrate why this may not be a general fact?
An example
When calculating the distance travelled as an integral of velocity with respect to time, the units are the same as when multiplying the two quantities.
$$ \int v \, \mathrm{d}t = s $$ $$ [\mathrm{m}/\mathrm{s}][\mathrm{s}] = [\mathrm{m}]$$
In thermodynamics, when calculating the entropy as a partial derivative of the internal energy with respect to temperature, the units are the same as when dividing the former quantity by the latter.
$$ \left(\frac{\partial{U}}{\partial{T}}\right)_{V,\mathbf{n}} = S $$ $$ \frac{[\mathrm{J}]}{[\mathrm{K}]} = \frac{[\mathrm{J}]}{[\mathrm{K}]}$$
Disclaimer
This is probably a rather trivial question, and a strongly suspect that the answer is yes based on the relationship between the operations. However, I would love to get a proper answer based on sound mathematical arguments. Any helpful suggestions or hints are greatly appreciated!