`f(x) = x(1 - 4/(x+3))` Find the derivative of the algebraic function.

Textbook Question

Chapter 2, 2.3 - Problem 27 - 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 function, hence you need to use the quotient rule, but first you need to find the common denominator within round parenthesis, such that:

`f(x) = x*(1 - 4/(x+3))`

`f(x) = x*((x+3-4)/(x+3))`

`f(x) = x*((x-1)/(x+3))`

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

Differentiating with respect to x yields:

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

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

`f'(x) = (2x^2 + 6x - x - 3 - x^2 + x)/((x+3)^2)`

Combining like terms yields:

`f'(x) = (x^2 + 6x - 3)/((x+3)^2)`

Hence, evaluating the derivative of the function, yields `f'(x) = (x^2 + 6x - 3)/((x+3)^2).`

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