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Differentiate $ f $ and find the domain of $ f. $

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

$f^{\prime}(x)$ $=\frac{2 x-1-(x-1) \ln (x-1)}{(x-1)[1-\ln (x-1)]^{2}}$$\operatorname{Dom}(f)$ $=\left\{x | x>1 \text { and } x-1 \neq e^{1}\right\}=\{x | x>1 \text { and } x \neq 1+e\}=(1,1+e) \cup(1+e, \infty)$

01:51

Frank L.

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

Missouri State University

Campbell University

Baylor University

University of Michigan - Ann Arbor

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in this fellow we're gonna function in west aren't every divine the domain of dysfunction. Let's start from the narrative at prom effects were gonna his portion also election their job function the numerator multiplied by function denominator minus function in the numerator multiplied by defamatory bunch of India. No matter which is negative for about it by function in the denominator squared. All right, let's, uh, since we have explain this morning and denominated here, let's divide and multiply in a first term, uh, under new ranger by X minus one. So we have exploit us for brightness explained swarm of connection. Like explain this one. Well, thanks. Divided by X minus one times one minus. Next lawyer Explain this one. It's where All right. We can then right to preserve it in a family of ice as treks minus one minus Explain its full in times next year longer explain this. Would you write it by x minus? Well, in times, uh, one minus next log. Excellent. Swollen for all right. Now we're gonna decide on the domain. We know that first of all, we have natural light of explain this one in the dino man right here. you know that? Explain this. One can not be zero. It actually should be greater than zero. Right. So to me, that eggs should be greater than or broker. This is the first condition. What else, don't you? We also know that natural law Oh x minus form should be, uh, cannot be warm. So this cannot be a war. Why? Because this is one. We would get zero in the denominator and that would make dysfunction of bikes. Undefended says nature. Lagerback's minus one Cannot people with me that experts warn people he x cannot be going t plus one. So then we're gonna write the domain. Yes. First part is we know that extra be great. Once it will be between four plus four. Union, get bus form to father to

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