The NSW Education Standards Authority Mathematics Reference Sheet shows the Product rule as
and the Quotient rule as
These forms of the rules have been used many times in various textbooks and no doubt taught to most calculus students.
I don't understand why these rules are continually presented this way unless the authors want to confuse students.
Why woukd you reverse the order of the term going from the Product to the Quotient rule? That doesn't make sense.
It has been my experience that students are invariably taught to write down u and v, calculate du/dx and dv/dx, write the appropriate rule and then substitute the relevant expressions.
This is very inefficient. However it is a consequence of using the rules in the form they are presented above.
There is a much better way!!
I teach my students to use the Product rule in this way:
Given y = uv then y' = u'v + uv' (where y' mean dy/dx)
and given y = u/v then y' = (u'v - uv')/v2
The order of the functions u and v remains the same in each term of the Product rule and we differentiate each function in turn, keeping the other unchanged. The only difference between the Product and Quotient rules is that the addition becomes a subtraction and we divide by v2. It should be this way because the Quotient rule comes from applying the Product rule to u/v when written in the form uv-1.
Another advantage of using these forms is that it is much more efficient. All you need to do is write the rule then write the relevant expression directly underneath.
which can then be tidied up. .
Finally, if we have a product of three functions, y = uvw then y' = u'vw + uv'w + uvw' ie differentiate each function in turn, leaving the others unchanged. This is just an extension of the rule for two functions.