`f(x) = (x^4)(1 - 2/(x+1))` Find the derivative of the algebraic function.

Textbook Question

Chapter 2, 2.3 - Problem 28 - 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 find derivative of the function using the product rule:

`f'(x)= (x^4)'*((x+1-2)/(x+1)) + (x^4)*((x-1)/(x+1))'`

You need to use the quotient rule to differentiate ((x-1)/(x+1)):

`f'(x)= (4x^3)*((x-1)/(x+1)) + (x^4)*((x-1)'*(x+1) - (x-1)*(x+1)')/((x+1)^2)`

`f'(x)= (4x^3)*((x-1)/(x+1))+ (x^4)*(x+1 - x + 1)/((x+1)^2)`

Reducing like terms yields:

`f'(x)= (4x^3)*((x-1)/(x+1))+ 2*(x^4)/((x+1)^2)`

You need to factor out `(2x^3)/(x+1):`

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

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

Hence, evaluating the derivative of the function, yields `f'(x)= (2x^3)/(x+1)*((2x^2 + x - 2)/(x+1)).`

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