



 
Shafira, The "/" in "d/dx" is not really a fraction, but it behaves enough like one (eg, in the chain rule) that it is convenient to pretend it is notationally. We still have to be aware that it is not division and that we can't do everything that we could it if were. In the same way, in the alternative "operator notation" notation Dy [differentiating with respect to an agreedupon but unstated variable] the D acts a little bit like a multiplication, so we can write things like $(D^2+ 2D+3)y$ meaning $y" + 2y' + 3y.$ This is useful in differential equations. But we must be very careful here; for instance, $Dy/Dz$ is not equal to $y/z.$ But in the symbol "dx" the d is not even an operator and we cannot usefully pretend that it is being multiplied by the x. The differential "dx" must be treated as an unbreakable unit; so we square it, not its component letters. In the same way, we never write $sin^2(x)$ Good Hunting!
 


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