# 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.

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

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

`(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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