Why can you move the derivatives as like the denominator and numerators of a division
for example. if a = dv/dt then why is dv = a * dt a correct representation
Because the chain rule results in a product, and the manipulation you describe is a way of handling substitutions that enable us to work backwards through the chain rule. Hence the utility of the fractional notation. But (at least at the level of ordinary single variable calculus) we can always (and often prefer to) use other notations. You may be interested in a related discussion here: http://www.mathhelpforum.com/math-he...rmulas-dx.html
Don't integrate - balloontegrate!
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