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If I have some calculation $X=v\cdot\delta$ and I want to calculate $\frac{\partial}{\partial v}X$, the result according to Mathematica is $\delta$. Now, I want to define $\delta$ as a function of $v$ but without defining the exact formula because I do not have that as a whole but numerically. Thus, I want the result to be $$\frac{\partial}{\partial v}X=\delta+v\frac{\partial \delta}{\partial v}$$ which is just the application of the product formula for derivatives. How to set some kind of a "flag" or "marker" to prevent Mathematica from deriving $\delta$ after $v$ and saying that the result is always $0$ because the variables are named differently?

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You can use Dt:

Dt[v δ, v]

δ + v Dt[δ, v]

Or D with the option NonConstants:

D[v δ, v, NonConstants -> {δ}]

δ + v D[δ, v, NonConstants -> {δ}]

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