Section 3.5 : Derivatives of Trig Functions

For problems 1 – 6 evaluate the given limit.

1. \(\displaystyle \mathop {\lim }\limits_{t \to \,0} \frac{{3t}}{{\sin \left( t \right)}}\)
2. \(\displaystyle \mathop {\lim }\limits_{w \to \,0} \frac{{\sin \left( {9w} \right)}}{{10w}}\)
3. \(\displaystyle \mathop {\lim }\limits_{\theta \to \,0} \frac{{\sin \left( {2\theta } \right)}}{{\sin \left( {17\theta } \right)}}\)
4. \(\displaystyle \mathop {\lim }\limits_{x \to \, - 4} \frac{{\sin \left( {x + 4} \right)}}{{3x + 12}}\)
5. \(\displaystyle \mathop {\lim }\limits_{x \to \,0} \frac{{\cos \left( x \right) - 1}}{{9x}}\)
6. \(\displaystyle \mathop {\lim }\limits_{z \to \,0} \frac{{\cos \left( {8z} \right) - 1}}{{2z}}\)

For problems 7 – 16 differentiate the given function.

1. \(h\left( x \right) = {x^4} - 9\sin \left( x \right) + 2\tan \left( x \right)\)
2. \(g\left( t \right) = 8\sec \left( t \right) + \cos \left( t \right) - 4\csc \left( t \right)\)
3. \(y = 6\cot \left( w \right) - 8\cos \left( w \right) + 9\)
4. \(f\left( x \right) = 8\sec \left( x \right)\csc \left( x \right)\)
5. \(h\left( t \right) = 8 - {t^9}\tan \left( t \right)\)
6. \(R\left( x \right) = 6\,\sqrt[5]{{{x^2}}} + 8x\sin \left( x \right)\)
7. \(\displaystyle h\left( z \right) = 3z - \frac{{\cos \left( z \right)}}{{{z^3}}}\)
8. \(\displaystyle Y\left( x \right) = \frac{{1 + \cos \left( x \right)}}{{1 - \sin \left( x \right)}}\)
9. \(\displaystyle f\left( w \right) = 3w - \frac{{\sec \left( w \right)}}{{1 + 9\tan \left( w \right)}}\)
10. \(\displaystyle g\left( t \right) = \frac{{t\cot \left( t \right)}}{{{t^2} + 1}}\)
11. Find the tangent line to \(f\left( x \right) = 2\tan \left( x \right) - 4x\) at \(x = 0\).
12. Find the tangent line to \(f\left( x \right) = x\sec \left( x \right)\) at \(x = 2\pi \).
13. Find the tangent line to \(f\left( x \right) = \cos \left( x \right) + \sec \left( x \right)\) at \(x = \pi \).
14. The position of an object is given by \(s\left( t \right) = 9\sin \left( t \right) + 2\cos \left( t \right) - 7\) determine all the points where the object is not changing.
15. The position of an object is given by \(s\left( t \right) = 8t + 10\sin \left( t \right)\) determine where in the interval \(\left[ {0,12} \right]\) the object is moving to the right and moving to the left.
16. Where in the range \(\left[ { - 6,6} \right]\) is the function \(f\left( z \right) = 3z - 8\cos \left( z \right)\) is increasing and decreasing.
17. Where in the range \(\left[ { - 3,5} \right]\) is the function \(R\left( w \right) = 7\cos \left( w \right) - \sin \left( w \right) + 3\) is increasing and decreasing.
18. Where in the range \(\left[ {0,10} \right]\) is the function \(h\left( t \right) = 9 - 15\sin \left( t \right)\) is increasing and decreasing.
19. Using the definition of the derivative prove that \(\frac{d}{{dx}}\left( {\cos \left( x \right)} \right) = - \sin \left( x \right)\).
20. Prove that \(\displaystyle \frac{d}{{dx}}\left( {\sec \left( x \right)} \right) = \sec \left( x \right)\tan \left( x \right)\).
21. Prove that \(\displaystyle \frac{d}{{dx}}\left( {\cot \left( x \right)} \right) = - {\csc ^2}\left( x \right)\).
22. Prove that \(\displaystyle \frac{d}{{dx}}\left( {\csc \left( x \right)} \right) = - \csc \left( x \right)\cot \left( x \right)\).