`f(x) = (2x + 7)/(x^2 + 4)` Use the Product Rule or the Quotient Rule to find the derivative of the function.

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Chapter 2, Review - Problem 34 - 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) = ((2x+7)'(x^2+4) - (2x+7)(x^2+4)')/((x^2+4)^2)`

`f'(x) = (2(x^2+4) - (2x+7)(2x))/((x^2+4)^2)`

`f'(x) = (2x^2 + 8 - 4x^2 - 14x)/((x^2+4)^2)`

Combining like terms yields:

`f'(x) = (-2x^2 - 14x + 8)/((x^2+4)^2)`

Hence, evaluating the derivative of the function, using the product rule, yields `f'(x) = (-2x^2 - 14x + 8)/((x^2+4)^2).`

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