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

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Problem 27 Medium Difficulty

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

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

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Video Transcript

In this problem were given a function that we're just to find derivative and the domain of this function. Let'S start from a derivative primefit we're going to use question to function. Derivative function in the numerator multiplied by function, denominator: minus functional, numerator, multiplied by derivative of the function in denominator, which is negative 4 over x, 11 divided by function in the denominator square, all right now, let's, since we have x ponent found in the denominator here, let's Divide and multiply the first term only numerator by x, minus 1, so we have x, minus 1 minus x minus 1 times natural lot x, minus 1 plus x, divided by x, minus 1 times 1 minus natural log of x minus 1 square. All right! We can then write the preservative f prime of x as 2 x, minus 1 minus x minus 1 times natural log of x, minus 1, divided by x, minus 1 times 1 minus, actually log of x minus 1 square. All right now we're going to decide on the domain. We know that, first of all, we have nental lot of x minus 1 in the denominator right here. We know that x, minus 1 can not be 0. It actually should be greater than 0 right. So it means that x should be greater than or this is the first condition. What else can we do? We also know that natural log of x, minus form should be, cannot be more so this cannot be equal to 1. Why? Because this is 1, we would get 0 in the denominator, and that would make the expansion of effects undefined. Since natural log of x, minus 1 cannot be 1. It means that x, minus 1 cannot be equal to e, so x cannot be equal to e plus 1 point. So then we're going to write the domain as first part is. We know that i should be greater than 1, so it will be between a and e plus 1 union plus 1 to positive infinity.

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Calculus: Early Transcendentals

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