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Numerade Educator

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Problem 5 Easy Difficulty

Differentiate the function.

$ f(x) = \ln \frac {l}{x} $

Answer

$f^{\prime}(x)=-\frac{1}{x}$

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Video Transcript

So for this problem, we want to use some properties of logarithms before we actually take the derivative. It's often easier to simplify the law, get into its simplest form Before we start. I'm taking derivatives. So what we have here is the natural log. Half of ax equals a natural log of one over X. So when we have this, we will a za results have. This is the same thing as the natural log of X to the negative one. But that mind, we know that this is the same thing as negative one. I'm the natural log of acts or just the negative natural Lagerback's but just using properties of logs that we already know from before. But now we can differentiate this, and it's much simpler. When we take the derivative of this, we can pull out the negative because it's just a constant. It's a negative one, essentially. So we know that f prime of X will end up equaling a negative one over X. And just to prove that we did this properly, we can take f prime of X When we see that the graph matches particularly right here, thgraf will match again because we manipulated the logarithms log rhythm of it. It's going to change on Lee. It will only match half of it. But since we did the proper steps, we see that this will be the derivative right here on this. Partially matches what? We're looking for this blue portion right here.