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Chain rule practice problems

Derivatives of compositions: differentiate the outside, keep the inside, multiply by the inside's derivative. 297 problems with worked solutions; in 281 of them every equation is proved by a computer algebra system.

Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(x \right)} \right)} \).
Problem 2.1medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \cos{\left(4 x + 2 \right)}}{4} \).
Problem 2.2medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(4 x \right)} \right)}}{4} \).
Problem 2.5medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(3 x - 1 \right)}}{3} \).
Problem 2.17medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \ln{\left(\ln{\left(x \right)} \right)} \).
Problem 2.19medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \sin{\left(4 x \right)}}{4} \).
Problem 2.20medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \tan{\left(3 x \right)} \).
Problem 2.26medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(2 x - 3 \right)} \).
Problem 2.29medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{e^{2 x - 1}}{2} \).
Problem 2.31medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(x - 3 \right)} \).
Problem 2.35medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{e^{3 x - 1}}{3} \).
Problem 2.38medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \cos{\left(2 x - 3 \right)}}{2} \).
Problem 2.39medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(2 x + 2 \right)}}{2} \).
Problem 2.44medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{3 x} \).
Problem 2.45medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(3 x + 1 \right)}}{3} \).
Problem 2.52medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 \sin{\left(x - 3 \right)} \).
Problem 2.53medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(2 x - 3 \right)}}{2} \).
Problem 2.59medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 \sin{\left(3 x + 2 \right)}}{3} \).
Problem 2.62medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(5 x - 3 \right)}}{5} \).
Problem 2.64medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 \sin{\left(5 x + 1 \right)}}{5} \).
Problem 2.67medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x + 2 \right)}}{2} \).
Problem 2.70medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{e^{4 x + 2}}{4} \).
Problem 2.73medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{2 x} \).
Problem 2.74medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\ln{\left(4 x \right)} \right)}}{4} \).
Problem 2.76medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(3 x - 1 \right)} \).
Problem 2.79medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(5 x \right)} \right)}}{5} \).
Problem 2.82medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{5 x} \).
Problem 2.91medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(5 x \right)}}{5} \).
Problem 2.93medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = - 3 \cos{\left(x - 3 \right)} \).
Problem 2.95medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 \ln{\left(\ln{\left(x + 2 \right)} \right)} \).
Problem 2.105medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \cos{\left(x - 1 \right)} \).
Problem 2.106medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(3 x \right)}}{3} \).
Problem 2.110medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 e^{2 x + 2}}{2} \).
Problem 2.113medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \ln{\left(x + 1 \right)} \).
Problem 2.115medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(3 x \right)}}{3} \).
Problem 2.117medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x + 1 \right)}}{2} \).
Problem 2.120medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.121medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{e^{4 x}}{4} \).
Problem 2.124medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.125medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x \right)}}{2} \).
Problem 2.128medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(2 x \right)}}{2} \).
Problem 2.130medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\tan{\left(4 x \right)}}{4} \).
Problem 2.137medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(2 x - 1 \right)} \right)}}{2} \).
Problem 2.138medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = e^{x + 1} \).
Problem 2.142medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3} \).
Problem 2.152medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(2 x + 1 \right)} \right)}}{2} \).
Problem 2.153medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 \ln{\left(x - 3 \right)} \).
Problem 2.160medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{2 \cos{\left(3 x - 3 \right)}}{3} \).
Problem 2.161medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(2 x - 3 \right)} \right)}}{2} \).
Problem 2.167medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(2 x + 1 \right)} \).
Problem 2.172medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{2 x - 3} \).
Problem 2.174medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x + 2 \right)} \).
Problem 2.177medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\ln{\left(2 x + 1 \right)} \right)}}{2} \).
Problem 2.178medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(2 x - 1 \right)}}{2} \).
Problem 2.184medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(2 x - 1 \right)}}{2} \).
Problem 2.194medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 e^{x - 3} \).
Problem 2.195medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(2 x \right)} \right)}}{2} \).
Problem 2.200medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(4 x - 1 \right)}}{4} \).
Problem 2.209medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \sin{\left(x - 1 \right)} \).
Problem 2.210medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 e^{2 x}}{2} \).
Problem 2.212medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(4 x \right)}}{4} \).
Problem 2.219medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 e^{x + 2} \).
Problem 2.221medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \sin{\left(2 x \right)}}{2} \).
Problem 2.224medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \ln{\left(\ln{\left(x - 3 \right)} \right)} \).
Problem 2.225medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{e^{4 x}}{4} \).
Problem 2.229medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 e^{3 x}}{3} \).
Problem 2.232medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(3 x - 1 \right)}}{3} \).
Problem 2.233medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(5 x - 1 \right)}}{5} \).
Problem 2.235medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.239medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(4 x - 3 \right)}}{4} \).
Problem 2.241medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x - 3 \right)}}{2} \).
Problem 2.243medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(2 x - 1 \right)}}{2} \).
Problem 2.246medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(2 x + 2 \right)} \).
Problem 2.249medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \cos{\left(2 x \right)} \).
Problem 2.252medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 e^{5 x}}{5} \).
Problem 2.253medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x - 1 \right)}}{2} \).
Problem 2.257medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{x + 2} \).
Problem 2.259medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(x + 1 \right)} \).
Problem 2.262medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 e^{2 x - 1}}{2} \).
Problem 2.266medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 \sin{\left(5 x - 3 \right)}}{5} \).
Problem 2.267medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(2 x - 3 \right)}}{2} \).
Problem 2.268medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \sin{\left(3 x \right)}}{3} \).
Problem 2.271medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(5 x + 2 \right)} \).
Problem 2.275medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\tan{\left(5 x \right)}}{5} \).
Problem 2.276medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(5 x + 1 \right)}}{5} \).
Problem 2.279medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x + 2 \right)}}{2} \).
Problem 2.280medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(2 x \right)} \right)}}{2} \).
Problem 2.285medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(5 x \right)} \right)} \).
Problem 2.287medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(4 x \right)}}{4} \).
Problem 2.291medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = 2 \sin{\left(x + 2 \right)} \).
Problem 2.300medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(x - 3 \right)} \right)} \).
Problem 2.301medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(2 x + 1 \right)}}{2} \).
Problem 2.302medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{e^{2 x + 2}}{2} \).
Problem 2.303medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \cos{\left(4 x \right)}}{4} \).
Problem 2.305medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(3 x - 3 \right)}}{3} \).
Problem 2.306medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(5 x + 2 \right)}}{5} \).
Problem 2.311medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \sqrt{\left(x - 3\right)^{2} + 1} \).
Problem 2.341medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \operatorname{asin}{\left(2 x - 1 \right)}}{2} \).
Problem 2.398medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(2 x \right)}}{2} \).
Problem 2.507medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{2 \operatorname{asin}{\left(5 x \right)}}{5} \).
Problem 2.512medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(4 x \right)}}{2} \).
Problem 2.541medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(4 x \right)}}{4} \).
Problem 2.557medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(2 x - 1 \right)} \).
Problem 2.560medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{\left(4 x - 1\right)^{2} + 1}}{4} \).
Problem 2.563medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(2 x \right)} \).
Problem 2.566medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \).
Problem 2.574medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(x - 1 \right)} \).
Problem 2.598medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{\left(3 x + 1\right)^{2} + 1}}{3} \).
Problem 2.629medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(5 x - 3\right)^{2} + 1}}{5} \).
Problem 2.675medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \).
Problem 2.680medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \sqrt{\left(5 x - 3\right)^{2} + 1} \).
Problem 2.702medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \).
Problem 2.712medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \).
Problem 2.3hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \).
Problem 2.4hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x \ln{\left(4 x + 2 \right)} - x + \frac{\ln{\left(2 x + 1 \right)}}{2} \).
Problem 2.6hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{x}{x + 1} \right)} \).
Problem 2.8hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(- \cot{\left(x \right)} + \csc{\left(x \right)} \right)} \).
Problem 2.9hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) \).
Problem 2.13hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{3 \ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \).
Problem 2.14hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(4 x - 1\right) e^{4 x}}{2} \).
Problem 2.15hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(2 x - 3 \right)} + \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).
Problem 2.18hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\sin{\left(x \right)} \right)} \).
Problem 2.21hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\sec{\left(x \right)} \right)} \).
Problem 2.23hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \).
Problem 2.24hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(5 x \right)} \right)}}{5} \).
Problem 2.27hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \).
Problem 2.28hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8} \).
Problem 2.32hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \).
Problem 2.33hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} \).
Problem 2.34hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{20} \).
Problem 2.41hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2} - \ln{\left(\tan{\left(x + 1 \right)} \right)} \).
Problem 2.43hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)} + 2 \ln{\left(\tan{\left(x - 1 \right)} \right)} \).
Problem 2.46hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \).
Problem 2.48hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{x}{2} + \frac{\ln{\left(2 x - 1 \right)}}{4} \).
Problem 2.49hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(2 x - 1 \right)} - 5 x - \frac{5 \ln{\left(2 x - 1 \right)}}{2} \).
Problem 2.51hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(1 - \ln{\left(4 x \right)}\right) \).
Problem 2.54hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \ln{\left(\cos{\left(5 x \right)} \right)} \).
Problem 2.56hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} \).
Problem 2.58hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \).
Problem 2.60hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x + 4 \right)}}{8} \).
Problem 2.61hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \).
Problem 2.63hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(5 x + 1 \right)} + 1 \right)}}{10} \).
Problem 2.65hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x - 2 \right)}}{20} \).
Problem 2.69hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{6} \).
Problem 2.71hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(2 x \right)} - 1\right) \).
Problem 2.72hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x + 4 \right)}}{12} \).
Problem 2.77hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\cot{\left(2 x + 2 \right)} + \csc{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.80hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x e^{2 x + 1} \).
Problem 2.81hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} - \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \).
Problem 2.83hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{4} - \frac{\ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{4} \).
Problem 2.84hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} \).
Problem 2.88hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x e^{2 x + 1} \).
Problem 2.89hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x \ln{\left(x - 3 \right)} - x - 3 \ln{\left(x - 3 \right)} \).
Problem 2.90hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 x e^{4 x + 1} \).
Problem 2.94hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{3 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \).
Problem 2.97hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)}}{3} \).
Problem 2.99hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{6} \).
Problem 2.103hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2} \).
Problem 2.104hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{2} \).
Problem 2.108hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \).
Problem 2.109hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(10 x \right)}}{10} \).
Problem 2.111hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x - 1 \right)} \right)}}{2} \).
Problem 2.112hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x - 2 \right)}}{4} \).
Problem 2.116hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{5} \).
Problem 2.118hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(6 x \right)}}{6} \).
Problem 2.119hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} \).
Problem 2.122hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.126hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \).
Problem 2.127hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)} \right)}}{4} \).
Problem 2.129hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \left(x - 4\right) e^{x - 3} \).
Problem 2.131hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{4} \).
Problem 2.132hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{2 x} \cos{\left(2 x + \frac{\pi}{4} \right)}}{4} \).
Problem 2.133hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x \ln{\left(4 x - 3 \right)} + x + \frac{3 \ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.134hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)} \).
Problem 2.135hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(2 x + 2 \right)} - 2 x + 2 \ln{\left(x + 1 \right)} \).
Problem 2.136hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x + 4 \right)}}{8} \).
Problem 2.139hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{12} \).
Problem 2.140hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{6} \).
Problem 2.144hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{3} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{3} \).
Problem 2.147hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(2 x + \frac{2}{5}\right) e^{5 x + 2} \).
Problem 2.148hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(5 x + 1\right) e^{5 x + 2} \).
Problem 2.149hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)} \right)}}{3} \).
Problem 2.150hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x \left(x - 2\right) \ln{\left(x - 1 \right)} + x + \ln{\left(x - 1 \right)} \).
Problem 2.154hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(3 x - 1 \right)} + 1 \right)}}{6} \).
Problem 2.156hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x + 2 \right)}}{2} \).
Problem 2.157hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} - \ln{\left(\tan{\left(x - 1 \right)} \right)} \).
Problem 2.158hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(4 x \right)} - 1 \right)}}{8} + \frac{\ln{\left(\sin{\left(4 x \right)} + 1 \right)}}{8} \).
Problem 2.159hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(5 x \right)} - 1\right) \).
Problem 2.162hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(1 - \ln{\left(3 x \right)}\right) \).
Problem 2.164hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\frac{1}{2} - x\right) e^{2 x} \).
Problem 2.166hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\tan^{2}{\left(5 x - 3 \right)} + 1 \right)}}{10} \).
Problem 2.168hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \).
Problem 2.170hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(3 x \right)} \right)}}{3} \).
Problem 2.173hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \).
Problem 2.175hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(2 x - 1\right) e^{2 x} \).
Problem 2.179hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x \right)}}{8} \).
Problem 2.181hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{1}{2}\right) e^{2 x} \).
Problem 2.182hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(2 x + 1 \right)} + 1 \right)}}{4} - \frac{\ln{\left(\tan{\left(2 x + 1 \right)} \right)}}{2} \).
Problem 2.183hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{5} \).
Problem 2.185hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x - 6 \right)}}{8} \).
Problem 2.186hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2} \).
Problem 2.187hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}}{8} \).
Problem 2.188hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2} \).
Problem 2.190hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \).
Problem 2.191hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(2 x - 1 \right)} + 1 \right)}}{4} \).
Problem 2.192hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x - 2 \right)}}{20} \).
Problem 2.193hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{1}{4}\right) e^{4 x} \).
Problem 2.196hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - 3 \ln{\left(\cos{\left(x \right)} \right)} \).
Problem 2.197hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(3 x \right)} \right)}}{3} \).
Problem 2.203hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(3 x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(3 x \right)} + 1 \right)}}{2} \).
Problem 2.204hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x \ln{\left(4 x - 3 \right)} - x - \frac{3 \ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.205hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{2 \ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \).
Problem 2.206hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \).
Problem 2.207hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).
Problem 2.211hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(5 x - \frac{20}{3}\right) e^{3 x - 3} \).
Problem 2.213hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(5 x \right)} - 1 \right)}}{10} + \frac{\ln{\left(\cos{\left(5 x \right)} + 1 \right)}}{10} \).
Problem 2.216hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \).
Problem 2.217hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x + 2 \right)}}{8} \).
Problem 2.218hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{5 \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \).
Problem 2.220hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \).
Problem 2.223hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{5 \ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \).
Problem 2.226hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x - 6 \right)}}{20} \).
Problem 2.228hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \ln{\left(5 x + 2 \right)} - x + \frac{2 \ln{\left(5 x + 2 \right)}}{5} \).
Problem 2.230hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 x e^{x + 1} \).
Problem 2.231hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{2} \).
Problem 2.234hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(3 x \right)} \right)}}{3} \).
Problem 2.237hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x^{2} \left(2 \ln{\left(5 x \right)} - 1\right)}{4} \).
Problem 2.238hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} + \frac{\ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \).
Problem 2.245hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)} \).
Problem 2.247hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2} \).
Problem 2.251hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \ln{\left(3 x + 2 \right)} - x + \frac{2 \ln{\left(3 x + 2 \right)}}{3} \).
Problem 2.254hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \ln{\left(3 x - 3 \right)} - x - \ln{\left(x - 1 \right)} \).
Problem 2.255hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(2 x + 1\right) e^{2 x + 2} \).
Problem 2.258hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{3 \ln{\left(\cos{\left(x \right)} + 1 \right)}}{2} \).
Problem 2.260hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(x - 3 \right)} - 5 x - 15 \ln{\left(x - 3 \right)} \).
Problem 2.261hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(4 x \right)} \right)}}{4} \).
Problem 2.263hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{\ln{\left(\cos{\left(x \right)} + 1 \right)}}{2} \).
Problem 2.264hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(3 x \right)} - 1\right) \).
Problem 2.265hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)} \).
Problem 2.269hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{12} \).
Problem 2.270hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(5 x \right)} \right)}}{5} \).
Problem 2.272hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x^{2} \sec{\left(x \right)} \right)} \).
Problem 2.273hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.277hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x \right)}}{4} \).
Problem 2.281hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\cos{\left(4 x \right)} \right)}}{4} \).
Problem 2.282hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(1 - x\right) e^{4 x - 3} \).
Problem 2.283hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} + \frac{3 \ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \).
Problem 2.284hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{2 x} \cos{\left(2 x + \frac{\pi}{4} \right)}}{4} \).
Problem 2.288hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - 3 \ln{\left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)} \right)} \).
Problem 2.289hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(10 \ln{\left(4 x \right)} - 5\right) \).
Problem 2.290hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x e^{4 x + 1} \).
Problem 2.292hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(5 x + \frac{5}{2}\right) e^{2 x + 2} \).
Problem 2.293hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(4 x - 3 \right)} - 5 x - \frac{15 \ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.294hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(x + 2 \right)} - 3 x + 6 \ln{\left(x + 2 \right)} \).
Problem 2.296hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \).
Problem 2.298hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2} + 5 \ln{\left(\tan{\left(x + 1 \right)} \right)} \).
Problem 2.299hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(4 x \right)} \right)}}{4} \).
Problem 2.304hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} - \ln{\left(\tan{\left(x - 3 \right)} \right)} \).
Problem 2.308hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - 5 \ln{\left(\cos{\left(x \right)} \right)} \).
Problem 2.309hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(3 x - 3 \right)}}{3 x - 2} \).
Problem 2.312hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(4 x + 2 \right)}}{4 x + 2} \).
Problem 2.313hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\cot{\left(2 x + 1 \right)} + \csc{\left(2 x + 1 \right)} \right)}}{2} \).
Problem 2.318hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} \).
Problem 2.319hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{x}{\sqrt{1 - x^{2}}} \right)} \).
Problem 2.348hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{9 x^{2} - 1}}{\sqrt{1 - 9 x^{2}}} \right)} \).
Problem 2.349hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{x^{2} - 1}}{\sqrt{1 - x^{2}}} \right)} \).
Problem 2.350hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = 2 x + \ln{\left(x^{2} \right)} + 6 \).
Problem 2.351hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \).
Problem 2.358hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} \).
Problem 2.399hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x - 2 \right)}}{2} \).
Problem 2.455hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x + 2 \right)}}{8} \).
Problem 2.467hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x - 6 \right)}}{8} \).
Problem 2.469hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(4 x + 2 \right)}}{4} \).
Problem 2.525hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)} \right)} \).
Problem 2.530hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{x + 1}}{\cos{\left(x + 1 \right)}} \right)} \).
Problem 2.531hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{4 x^{2} - 1}}{2 x} \right)} \).
Problem 2.532hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - x + \ln{\left(\frac{\sqrt{x^{2} - 1}}{\sqrt{1 - x^{2}}} \right)} \).
Problem 2.533hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(\frac{\sqrt{1 - 4 x^{2}}}{\sqrt{4 x^{2} - 1}} \right)} \).
Problem 2.534hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\cot{\left(2 x - 1 \right)} \right)} \).
Problem 2.535hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(4 x^{2} e^{2 x} \right)} \).
Problem 2.536hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x \right)}}{2} \).
Problem 2.543hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x - 6 \right)}}{20} \).
Problem 2.550hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x + 4 \right)}}{8} \).
Problem 2.552hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x \right)}}{8} \).
Problem 2.606hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x - 2 \right)}}{8} \).
Problem 2.611hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x + 4 \right)}}{8} \).
Problem 2.619hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x - 2 \right)}}{8} \).
Problem 2.673hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12} \).
Problem 2.684hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x + 2 \right)}}{8} \).
Problem 2.691hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x \right)}}{8} \).
Problem 2.692hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(6 x + 4 \right)}}{12} \).
Problem 2.696hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x \right)}}{4} \).
Problem 2.700hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x + 4 \right)}}{4} \).
Problem 2.718hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{1 - x^{2}}}{\sqrt{x^{2} - 1}} \right)} \).
Problem 2.721hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{2 x}}{2 x} \right)} \).
Problem 2.722hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \).
Problem 2.723hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(2 \sqrt{x^{2} + 1} \right)} \).
Problem 2.724hard✓ Every equation proved