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Derivatives of logarithms practice problems

ln x and ln of anything, where the chain rule gives u′/u. 50 problems with worked solutions; in 50 of them every equation is proved by a computer algebra system.

Differentiate \( \displaystyle f(x) = 5 \ln{\left(x \right)} \).
Problem 2.107easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x \right)} \).
Problem 2.189easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x - 1 \right)} \).
Problem 2.335easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(2 x + 1 \right)}}{2} \).
Problem 2.401easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(5 x + 2 \right)}}{2} \).
Problem 2.416easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(3 x - 3 \right)}}{3} \).
Problem 2.509easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(3 x + 2 \right)}}{3} \).
Problem 2.695easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x - 3 \right)} \).
Problem 2.717easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \ln{\left(x + 2 \right)} \).
Problem 2.719easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = 3 x \left(\ln{\left(x \right)} - 1\right) \).
Problem 2.227medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(x \right)} - 1\right) \).
Problem 2.286medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(4 x \right)} - 1\right) \).
Problem 2.376medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(1 - \ln{\left(x \right)}\right) \).
Problem 2.482medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 x \left(\ln{\left(5 x \right)} - 1\right) \).
Problem 2.491medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(3 x - 1 \right)}}{3} \).
Problem 2.569medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(2 x \right)}^{2} \).
Problem 2.626medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x \left(\ln{\left(3 x \right)} - 1\right) \).
Problem 2.703medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(1 - \ln{\left(2 x \right)}\right) \).
Problem 2.720medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \ln{\left(x \right)} - x \).
Problem 2.10hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \).
Problem 2.215hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x^{2} \left(1 - 2 \ln{\left(x \right)}\right)}{4} \).
Problem 2.236hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(e^{2 x} + 1 \right)} \).
Problem 2.343hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \ln{\left(2 x + 2 \right)} - x + \ln{\left(x + 1 \right)} \).
Problem 2.354hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(16 x^{2} + 1 \right)} \).
Problem 2.370hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \sqrt{3 x - 3} \ln{\left(3 x - 3 \right)} \).
Problem 2.395hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(x - 1 \right)} - 3 x - 3 \ln{\left(x - 1 \right)} \).
Problem 2.407hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - 1\right)^{2} \ln{\left(x - 1 \right)} \).
Problem 2.419hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(e^{2 x - 1} + 1 \right)} \).
Problem 2.424hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\left(2 x + 2\right)^{2} + 1 \right)} \).
Problem 2.429hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(e^{x + 1} + 1 \right)} \).
Problem 2.435hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\sin{\left(2 x - 1 \right)} \right)} \).
Problem 2.440hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(2 x - 3 \right)} - 5 x - \frac{15 \ln{\left(2 x - 3 \right)}}{2} \).
Problem 2.447hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(2 x \right)}}{4 x^{2}} \).
Problem 2.456hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(e^{x - 1} + 1 \right)} \).
Problem 2.466hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{3 x - 1} \ln{\left(3 x - 1 \right)} \).
Problem 2.476hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{2} \sqrt{x} \ln{\left(2 x \right)} \).
Problem 2.485hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(5 x \right)}}{25 x^{2}} \).
Problem 2.499hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\left(x - 3\right)^{2} + 1 \right)} \).
Problem 2.508hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{x} \ln{\left(x \right)} \).
Problem 2.526hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \ln{\left(2 x - 3 \right)} - x - \frac{3 \ln{\left(2 x - 3 \right)}}{2} \).
Problem 2.544hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \ln{\left(x \right)} \).
Problem 2.561hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\left(5 x - 3\right)^{2} + 1 \right)} \).
Problem 2.587hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 4 x^{2} \ln{\left(2 x \right)} \).
Problem 2.600hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x + 2\right)^{2} \ln{\left(x + 2 \right)} \).
Problem 2.608hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(x - 3\right)^{2} \ln{\left(x - 3 \right)} \).
Problem 2.614hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(e^{5 x + 2} + 1 \right)} \).
Problem 2.618hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\left(4 x + 1\right)^{2} + 1 \right)} \).
Problem 2.634hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(4 x - 3 \right)}}{\left(4 x - 3\right)^{2}} \).
Problem 2.640hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x \ln{\left(5 x + 1 \right)} - x + \frac{\ln{\left(5 x + 1 \right)}}{5} \).
Problem 2.648hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(5 x + 1 \right)} - 5 x + \ln{\left(5 x + 1 \right)} \).
Problem 2.682hard✓ Nihil obstat