Use logarithmic differentiation to find dy/dx , If `y=((x+1)^(7)/(2x+1)^(5))^(1/2)` Express your answer in terms of x only .

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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`y= ((x+1)^7 /(2xc+1)^5)^(1/2)`

`==>ln y= ln ((x+1)^7 / (2x+1)^5)^(1/2)`

`==>ln y = (1/2) (ln (x+1)^7 - ln (2x+1)^5)`

`==> ln y= (1/2) (7ln(x+1) - 5ln(2x+1))`

`==> ln y= (7/2) ln(x+1) - (5/2) ln (2x+1)`

`==> (ln y)' = (7/2) (ln(x+1))' - (5/2) (ln(2x+1))'`

`==>(y')/y = (7/2) * 1/(x+1) - (5/2)*2/(2x+1)`

`==>(y')/y=7/(2x+2) - 5/(2c+1)`

`==> y' = y (7/(2x+2) - 5/(2x+1))`

`==> y' = ((x+1)^7 /(2x+1)^5)^(1/2) (7/(2x+2) - 5/(2x+1))`

`==> y'= (7(x+1)^(5/2) )/(2(2x+1)^(5/2)) - (5(x+1)^7)/(2x+1)^6`

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