Shortcut for Quotient Rule

Hi Guys,
I remember back in the days of Calc I learning that there was an easy way to take multiple derivatives of certain functions that needed repeated uses of the quotient rule. I was wondering if anybody remembered that trick.
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 is this like using the difference of squares to reduce the function into a product of factors that you then can use the product or quotient rule on: http://calculustricks.com/page/3/

 Quote by jedishrfu is this like using the difference of squares to reduce the function into a product of factors that you then can use the product or quotient rule on: http://calculustricks.com/page/3/
It involved the denominator and easily reducing something in the numerator instead of having to do all the distributing and addition to then reduce. Instead of squaring the denominator everytime you only increased it's power by one. Also it has a factorial pattern in there if i recall correctly.
Like here:

http://www.wolframalpha.com/input/?i...%29&lk=4&num=1
http://www.wolframalpha.com/input/?i...+arctan%28x%29
http://www.wolframalpha.com/input/?i...+arctan%28x%29
http://www.wolframalpha.com/input/?i...+arctan%28x%29
http://www.wolframalpha.com/input/?i...+arctan%28x%29

and another function
http://www.wolframalpha.com/input/?i...%2F%28x%2B1%29
http://www.wolframalpha.com/input/?i...%2F%28x%2B1%29
http://www.wolframalpha.com/input/?i...%2F%28x%2B1%29
http://www.wolframalpha.com/input/?i=3rd+derivative+x%2F%28x^3%2B1%29

Shortcut for Quotient Rule

okay prof nearing in his math methods book describes a method differentiating a constant that seems to do that:

its called parametric differentiation in section 1.2 of the pdf listed below:

http://www.physics.miami.edu/~nearin...asic_stuff.pdf

 Tags calculus, derivative, function, quotient rule, second derivative