`y = x/(sqrt(x^4 + 4))` Find the derivative of the function.

Textbook Question

Chapter 2, 2.4 - 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 quotient rule:

`y' = (x'*(sqrt (x^4+4)) - x*(sqrt (x^4+4))')/(x^4+4)`

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

`y' = ((sqrt (x^4+4)) - (4x^4)/(2sqrt (x^4+4)))/(x^4+4)`

`y' = ((sqrt (x^4+4)) - (2x^4)/(sqrt (x^4+4)))/(x^4+4)`

`y'= (2x^4 + 8- 4x^4)/((2sqrt (x^4+4))*(x^4+4))`

`y'= (8- 2x^4)/((2sqrt (x^4+4))*(x^4+4))`

Factoring out 2 yields:

`y'= 2*(4- x^4)/((2sqrt (x^4+4))*(x^4+4))`

`y'= (4- x^4)/((x^4+4)*sqrt (x^4+4))`

Hence, evaluating the derivative of the function, yields `y'= (4- x^4)/((x^4+4)*sqrt (x^4+4)).`

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