Logarithmic Differentiation. Use LOGARIThMIC DIFFERENTIATION to find dy/dx,  if y=(((x+1)^(7))/(2x+1)^(5))^(1/2)))....Express your asnwer interms of in x only& as a single completely reduced fraction with rational exponents that are all positive exponents...Please show all work.

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You need to differentiate the function using chain rule and quotient rule but you do not need to use logarithmic differentiation.


`(dy)/(dx) = (((x+1)^7)'(sqrt((2x+1)^5)) - ((x+1)^7)(sqrt((2x+1)^5))')/((sqrt((2x+1)^5))^2)`

`(dy)/(dx) = (7(x+1)^6sqrt((2x+1)^5) - ((x+1)^7)(1/(2sqrt(2x+1)^5))*5(2x+1)^4*2)/((2x+1)^5)`

`(dy)/(dx) = (7(x+1)^6sqrt((2x+1)^5) - ((x+1)^7)(5(2x+1)^4)/(sqrt(2x+1)^5))/((2x+1)^5)`

`(dy)/(dx) = (7(x+1)^6(2x+1)^2sqrt(2x+1) - ((x+1)^7)(5(2x+1)^4)/(sqrt(2x+1)^5))/((2x+1)^5)`

You need to factor out `(x+1)^6(2x+1)^2`  such that:

`(dy)/(dx) = ((x+1)^6(2x+1)^2(7(sqrt(2x+1)) - (5(x+1)(2x+1)^2))/(sqrt(2x+1)^5))/((2x+1)^5)`

Reducing by `(2x+1)^2`  yields:

`(dy)/(dx) = ((x+1)^6(7(sqrt(2x+1)) - ((5(x+1)(2x+1)^2))/(sqrt(2x+1)^5)))/((2x+1)^3)`

Hence, evaluating the derivative of the given function yields `(dy)/(dx) = ((x+1)^6(7(sqrt(2x+1)) - ((5(x+1)))/(sqrt(2x+1))))/((2x+1)^3).`

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