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Derivatives of trigonometric functions practice problems

sin, cos, tan, sec, csc and cot, alone and in combination. 76 problems with worked solutions; in 54 of them every equation is proved by a computer algebra system.

Differentiate \( \displaystyle f(x) = \tan{\left(x \right)} \).
Problem 2.36easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \tan{\left(x \right)} \).
Problem 2.66easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cos{\left(x \right)} \).
Problem 2.101easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \cos{\left(x \right)} \).
Problem 2.171easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(x \right)} \).
Problem 2.214easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 \sin{\left(x \right)} \).
Problem 2.297easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\tan{\left(4 x \right)}}{4} \).
Problem 2.345easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = \cos{\left(x - 1 \right)} \).
Problem 2.392easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(3 x - 3 \right)}}{3} \).
Problem 2.405easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cos{\left(x + 2 \right)} \).
Problem 2.433easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = \sin{\left(3 x - 3 \right)} \).
Problem 2.462easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(5 x \right)}}{5} \).
Problem 2.495easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(3 x \right)}}{3} \).
Problem 2.501easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \cos{\left(x + 2 \right)} \).
Problem 2.573easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \cos{\left(5 x - 1 \right)} \).
Problem 2.596easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \sin{\left(x + 1 \right)} \).
Problem 2.610easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = \sin{\left(x - 1 \right)} \).
Problem 2.623easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\tan{\left(2 x \right)}}{2} \).
Problem 2.630easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 \tan{\left(2 x \right)}}{2} \).
Problem 2.671easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = - 5 \cos{\left(x - 3 \right)} \).
Problem 2.688easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 \tan{\left(5 x \right)}}{5} \).
Problem 2.708easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cos^{3}{\left(2 x \right)} \).
Problem 2.373medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \cos{\left(5 x + 1 \right)}}{5} \).
Problem 2.388medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(2 x + 2 \right)}}{2} \).
Problem 2.393medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 \sin{\left(2 x - 3 \right)}}{2} \).
Problem 2.449medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \cos{\left(4 x - 1 \right)}}{4} \).
Problem 2.475medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(2 x + 1 \right)}}{2} \).
Problem 2.545medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(4 x - 3 \right)}}{4} \).
Problem 2.578medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(3 x + 2 \right)}}{3} \).
Problem 2.601medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 \cos{\left(2 x + 1 \right)}}{2} \).
Problem 2.641medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{2 \cos{\left(5 x - 1 \right)}}{5} \).
Problem 2.659medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(5 x + 2 \right)}}{5} \).
Problem 2.664medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \cos{\left(5 x + 2 \right)}}{5} \).
Problem 2.689medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(2 x - 3 \right)} + \sec{\left(2 x - 3 \right)} \right)}}{2} \).
Problem 2.352hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.360hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \tan{\left(5 x + 1 \right)} \sec^{2}{\left(5 x + 1 \right)} \).
Problem 2.363hard≈ Checked numerically
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 2.368hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)} \right)} \).
Problem 2.383hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} \).
Problem 2.411hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \).
Problem 2.415hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)} \right)} \).
Problem 2.423hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\cos{\left(4 x \right)} - 1 \right)}}{8} - \frac{5 \ln{\left(\cos{\left(4 x \right)} + 1 \right)}}{8} \).
Problem 2.431hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - 2 \ln{\left(\cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \).
Problem 2.454hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{2 \ln{\left(\cos{\left(3 x \right)} \right)}}{3} \).
Problem 2.464hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)} \right)}}{2} \).
Problem 2.468hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - 5 \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)} \).
Problem 2.474hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \).
Problem 2.479hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(5 x - 1 \right)} + \sec{\left(5 x - 1 \right)} \right)} \).
Problem 2.500hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)} \right)} \).
Problem 2.503hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \ln{\left(\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)} \right)} \).
Problem 2.505hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \).
Problem 2.523hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{3 \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \).
Problem 2.528hard✓ 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 2.538hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{3 \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \).
Problem 2.542hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(2 x + 1 \right)} + 1 \right)}}{4} \).
Problem 2.555hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)} \).
Problem 2.567hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(5 x + 2 \right)} + \csc{\left(5 x + 2 \right)} \right)} \).
Problem 2.579hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \).
Problem 2.580hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 \ln{\left(\sin{\left(x \right)} \right)} \).
Problem 2.590hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(2 x - 3 \right)} + \csc{\left(2 x - 3 \right)} \right)}}{2} \).
Problem 2.605hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(5 x - 1 \right)} + 1 \right)}}{10} + \frac{\ln{\left(\tan{\left(5 x - 1 \right)} \right)}}{5} \).
Problem 2.613hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(5 x - 1 \right)} + 1 \right)}}{2} \).
Problem 2.637hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{6} \).
Problem 2.657hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} \).
Problem 2.662hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} \).
Problem 2.665hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(4 x \right)} \right)}}{2} \).
Problem 2.666hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{2 \ln{\left(\cot{\left(3 x - 3 \right)} + \csc{\left(3 x - 3 \right)} \right)}}{3} \).
Problem 2.678hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \).
Problem 2.681hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(5 x - 1 \right)} + \sec{\left(5 x - 1 \right)} \right)}}{5} \).
Problem 2.690hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)} \right)} \).
Problem 2.699hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(3 x \right)} - 1 \right)}}{3} - \frac{\ln{\left(\cos{\left(3 x \right)} + 1 \right)}}{3} \).
Problem 2.701hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{2} + \ln{\left(\tan{\left(3 x + 2 \right)} \right)} \).
Problem 2.704hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(2 x + 1 \right)} + \csc{\left(2 x + 1 \right)} \right)}}{2} \).
Problem 2.706hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(3 x + 2 \right)} + \sec{\left(3 x + 2 \right)} \right)}}{3} \).
Problem 2.707hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \).
Problem 2.709hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(4 x + 2 \right)} + \csc{\left(4 x + 2 \right)} \right)}}{4} \).
Problem 2.714hard✓ Nihil obstat