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I know the limit definition of a derivative. And I read at the end that if the limit exist then we say that function is differentiable and this existing limit is denoted by $\dfrac{d}{dx}$ . so we say that $\dfrac{d}{dx}$ is a notation. But during my graduation , in many proof of some theorems we do a step as Say.....

$\dfrac{d(f(x))}{dx} = g(x)$

Then multiply both sides by $dx$ we get... $d(f(x))= g(x)d(x)$

My question is that how we do this step....

Then i read total derivative of a function and it help me little bit .but don't understand completely ...... We used $\dfrac{d}{dx}$ as a notation so how do above step .... Please help me to understand it....

Khosrotash
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saman kumar
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1 Answers1

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You are right, using $\frac{\mathrm df}{\mathrm d}$ like any rational fraction is just invalid. Doing this is often seen as a physisicists/engineers trick for "proving" something without diving to deep into the not so important subleties.

More surprising, this notation and its informal use as fraction can still be made rigorous with using infinitesimals like, e.g. in the hyperreal numbers. Of course, you have to take care of more subtelies now. You have to keep in mind that you always seperate the usual reals from the infinitesimals.

I'm not so familiar with the correct historical framework around the foundation of this notation, but I think the original inventor had something in mind like the infinitesimal interpretation. Before the rigorous definition of limits, mathematicians often used the fuzzy concept of infinitesimals anyway.

M. Winter
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