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Find the derivatives of the given functions.$$y=\frac{8 \ln (2 x+1)}{x}$$

$\frac{d y}{d x}=\frac{16 x-8(2 x+1) \ln (2 x+1)}{x^{2}(2 x+1)}$

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 5

Derivative of the Logarithmic Function

Derivatives

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Yeah we want to find your I. D. X. For the function Why is equal to eight? Natural algorithm of two X plus one over X. This is testing our knowledge of differentiation shortcuts particularly those related to transcendental functions like logarithmic functions as such. That's not relevant rules before solving. Yeah, on the left we have the rules were taking the derivative of logarithmic functions on the right. We have 123 respectively. The power rule product rule in chain rule. We can rewrite why ask eight Ln to expose one xnegative first which shows us that we're going to use the product rule to begin this derivative and then we'll use both power rule zero and the general So buy the product rule dy dx is 8/2 X plus one times two X negative first. This is the derivative of Ln two X plus one by real zero minus eight Ln two X plus one X negative second. This is the power rule on xnegative first. Giving these two expressions or rather simplifying, produces do I d x equals 16 X minus eight times direct with one time natural order them to expose one all over X squared times direct. This one we gave these two terms like denominator.

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