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Find $ y' $ and $ y". $

$ y = \ln (l + \ln x) $

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

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Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

in this problem, we are getting comfortable with taking 1st and 2nd derivatives using the different differentiation. Roles in this case will be using chain rules, quotient, rule and also product rule. So let's first review the function that were given forgiven. F of X equals the natural log of one plus the natural log of X. So let's first find the first derivative F prime of X f Prime of X is going to equal to d the natural log of one plus X over d. X. So we have to apply the chain role. We have a composition of functions here, so F Private X will be equal to 1/1, plus a natural log of X time zero plus one over X so F prime of X would be equal to one over X times one plus the natural log of X, and that is our first derivative. Now, for the second derivative, it's a little bit more complicated. We'll have to apply the quotient rule and also the product rule, because not only do we have a quotient here, but in the denominator is a product so f double prime of X will be zero times x times one plus the natural log of X plus one times this entire quantity. This is where we have to plug in the product rule. You have ex prime times one plus a natural log of X plus X times one plus natural log, Ex prime all over X square times one plus the natural log of X squared. So once you take those derivatives and we simplify a little bit, we'll get F double Prime of X equals one plus the natural log of X plus one all over X square, times one plus a natural log of X squared and then we could simplify it. We'll get F double prime of X equals two plus the natural log of X all over X squared times one plus the natural log of X squared. So I hope this problem help to understand a little bit more about how we can take 1st and 2nd derivatives involving log rhythmic functions and also how we can apply the different the differentiation rules, and sometimes we have to apply them multiple times