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### 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 6 – 10 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 $$\displaystyle \frac{d}{{dx}}\left( {\csc \left( x \right)} \right) = - \csc \left( x \right)\cot \left( x \right)$$.