Which derivative corresponds to F(x) = -4 ln|x| + 7 ln|x-1|?

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Multiple Choice

Which derivative corresponds to F(x) = -4 ln|x| + 7 ln|x-1|?

Explanation:
The key idea is differentiating logarithmic terms using the rule d/dx[ln|u|] = u'/u. For the first term, u = x, so d/dx[ln|x|] = 1/x, and with the coefficient -4, you get -4/x. For the second term, u = x − 1, so d/dx[ln|x−1|] = 1/(x−1), and with the coefficient 7, you get 7/(x−1). Add them together: -4/x + 7/(x−1). This is valid for x ≠ 0 and x ≠ 1, since those are where the original logarithms are defined.

The key idea is differentiating logarithmic terms using the rule d/dx[ln|u|] = u'/u. For the first term, u = x, so d/dx[ln|x|] = 1/x, and with the coefficient -4, you get -4/x. For the second term, u = x − 1, so d/dx[ln|x−1|] = 1/(x−1), and with the coefficient 7, you get 7/(x−1). Add them together: -4/x + 7/(x−1). This is valid for x ≠ 0 and x ≠ 1, since those are where the original logarithms are defined.

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