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