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Since the derivative of x^-1 is -x^-2, is it correct to then work it further to get the "best answer" as 1/-sqrt x and then rationalize the denominator by multiplying by 1 as in (1/-sqrt x) * (-Sqrt x/-Sqrt X) to get -Sqrt x / x ???

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that probably depends on your teacher? Rationalizing it is certainly not incorrect.
My instructor counts off for math grammar I never heard of before.
ask him/her then to be sure. In my class I could leave it as x^-1

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Other answers:

This is my first negative exponent in a derivative. Wondering if it needs to be left as it comes out of the rule, or further refined to rationalized denominator form.
I've differentiated TWICE! Yeah!!!
In my textbook they leave it as a negative exponent, but that might be because of space considerations. Seriously, just ask your teacher.
and congratulations!!!

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