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for example, y = s - t becomes dy = ds - dt

Is this a legal operation according to mathematicians - or is it a hand-waving operation that is only used in the physical sciences and lacks pure mathematical rigor? Can this operation be done on functions too? or just equations?

Also there are often times when she writes that we differentiate a function without specifying with respect to what variable - just differentiating a function period. For example (Boas 2nd edition page 155), in E&M the equation R = kl/r^2 for wire resistance. She says we can find dR/R by differentiating ln R = ln k + ln l - 2ln r which gives us

dR/R = dl/l - 2dr/r

So is this differentiating with respect to some variable that I am not aware of? Or is this taking a differential of the equation ln R = ln k + ln l - 2ln r ?

If the latter, would that mean that differentiating nonspecific to a certain variable is equivalent to taking the differential of an equation/function?

Or are there 3 distinct operations? Differentiating nonspecific, differentiating with respect to a certain variable, and finding the differential of an equation/function? Thanks in advance for any clarification you can provide