`f(x) = (x^2 + x - 1)/(x^2 - 1)` Use the Product Rule or the Quotient Rule to find the derivative of the function.

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Chapter 2, Review - Problem 33 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the derivative of the given function and since the function is a quotient of two polynomials, then you must use the quotient rule, such that:

`f'(x) = ((x^2+x-1)'(x^2-1) - (x^2+x-1)(x^2-1)')/((x^2-1)^2)`

`f'(x) = ((2x+1)(x^2-1) - (x^2+x-1)(2x))/((x^2-1)^2)`

`f'(x) = (2x^3 - 2x + x^2 - 1 - 2x^3 - 2x^2 + 2x)/((x^2-1)^2)`

Reducing like terms yields:

`f'(x) = (-x^2 - 1)/((x^2-1)^2)`

Hence, evaluating the derivative of the function, using the product rule, yields `f'(x) = (-x^2 - 1)/((x^2-1)^2).`

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