Differentiate the function. y=ln((5y+1)^2/(sqrt(y^2+1))

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You need to write the function y in terms of x such that:

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

You need to differentiate the function with respect to x, thus you need to use quotient rule and chain rule such that:

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

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

`(dy)/(dx) = (10ln(5x+1)*(sqrt(x^2+1))/(5x+1) - ln((5x+1)^2*x/(sqrt(x^2+1))))/(x^2 + 1)`

Hence, differentiating the function with respect to x yields `(dy)/(dx) = (10ln(5x+1)*(sqrt(x^2+1))/(5x+1) - ln((5x+1)^2*x/(sqrt(x^2+1))))/(x^2 + 1).`

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