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What is derivative of y = ln(1+(2/x))?

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yapayapa | Honors

Posted June 28, 2013 at 4:35 PM via web

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What is derivative of y = ln(1+(2/x))?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 28, 2013 at 4:49 PM (Answer #1)

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You need to evaluate the derivative of composite function `y = ln(1 + 2/x)` , hence, you need to use the chain rule, such that:

`(dy)/(dx) = (d(ln(1 + 2/x)))/(dx)`

`(dy)/(dx) = 1/(1 + 2/x)*(d(1 + 2/x))/(dx)`

`(dy)/(dx) = 1/(1 + 2/x)*(0 + (2'*x - 2*x')/(x^2))`

`(dy)/(dx) = 1/(1 + 2/x)*(-2/x^2) => (dy)/(dx) = (-2/x^2)/(1 + 2/x)`

`(dy)/(dx) = (-2/x^2)/((2 + x)/x)`

Reducing duplicate factors yields:

`(dy)/(dx) = (-2)/(x(2 + x))`

Hence, evaluating the derivative of the given composite function, using the chain rule, yields `(dy)/(dx) = (-2)/(x(2 + x)).`

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