How do you find the derivative of #y=(2x^-1-x^-2)/(3x^-1-4x^-2)#?

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Answer 1 · 2016-12-11T21:58:29

Note the domain and rewrite, then use the quotient rule.

#f(x) = y = (2x^-1-x^-2)/(3x^-1-4x^-2)#

Domain of #f# is all reals except #0#, and #4/3#.

Now use algebra to rewrite

#f(x) = ((2x^-1-x^-2)x^2)/((3x^-1-4x^-2)x^2)#

# = (2x-1)/(3x-4)#

Now use the quotient rule

#f'(x) = ((2)(3x-4)-(3)(2x-1))/(3x-4)#

# = (-5)/(3x-4)^2#

If we want to make sure the domain is clear, then multiply to write

#f'(x) = (-5x^-1)/((3x-4)^2x^-1)#