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### Section 3-9 : Chain Rule

For problems 1 – 51 differentiate the given function.

1. $$g\left( x \right) = {\left( {3 - 8x} \right)^{11}}$$
2. $$g\left( z \right) = \sqrt{{9{z^3}}}$$
3. $$h\left( t \right) = {\left( {9 + 2t - {t^3}} \right)^6}$$
4. $$y = \sqrt {{w^3} + 8{w^2}}$$
5. $$R\left( v \right) = {\left( {14{v^2} - 3v} \right)^{ - 2}}$$
6. $$\displaystyle H\left( w \right) = \frac{2}{{{{\left( {6 - 5w} \right)}^8}}}$$
7. $$f\left( x \right) = \sin \left( {4x + 7{x^4}} \right)$$
8. $$T\left( x \right) = \tan \left( {1 - 2{{\bf{e}}^x}} \right)$$
9. $$g\left( z \right) = \cos \left( {\sin \left( z \right) + {z^2}} \right)$$
10. $$h\left( u \right) = \sec \left( {{u^2} - u} \right)$$
11. $$y = \cot \left( {1 + \cot \left( x \right)} \right)$$
12. $$f\left( t \right) = {{\bf{e}}^{1 - {t^{\,2}}}}$$
13. $$J\left( z \right) = {{\bf{e}}^{12z - {z^{\,6}}}}$$
14. $$f\left( z \right) = {{\bf{e}}^{z + \ln \left( z \right)}}$$
15. $$B\left( x \right) = {7^{\cos \left( x \right)}}$$
16. $$z = {3^{{x^{\,2}} - 9x}}$$
17. $$R\left( z \right) = \ln \left( {6z + {{\bf{e}}^z}} \right)$$
18. $$h\left( w \right) = \ln \left( {{w^7} - {w^5} + {w^3} - w} \right)$$
19. $$g\left( t \right) = \ln \left( {1 - \csc \left( t \right)} \right)$$
20. $$f\left( v \right) = {\tan ^{ - 1}}\left( {3 - 2v} \right)$$
21. $$h\left( t \right) = {\sin ^{ - 1}}\left( {9t} \right)$$
22. $$A\left( t \right) = \cos \left( t \right) - \sqrt{{1 - \sin \left( t \right)}}$$
23. $$H\left( z \right) = \ln \left( {6z} \right) - 4\sec \left( z \right)$$
24. $$f\left( x \right) = {\tan ^4}\left( x \right) + \tan \left( {{x^4}} \right)$$
25. $$f\left( u \right) = {{\bf{e}}^{4u}} - 6{{\bf{e}}^{ - u}} + 7{{\bf{e}}^{{u^{\,2}} - 8u}}$$
26. $$g\left( z \right) = {\sec ^8}\left( z \right) + \sec \left( {{z^8}} \right)$$
27. $$k\left( w \right) = {\left( {{w^4} - 1} \right)^5} + \sqrt {2 + 9w}$$
28. $$h\left( x \right) = \sqrt{{{x^2} - 5x + 1}} + {\left( {9x + 4} \right)^{ - 7}}$$
29. $$T\left( x \right) = {\left( {2{x^3} - 1} \right)^5}{\left( {5 - 3x} \right)^4}$$
30. $$w = \left( {{z^2} + 4z} \right)\sin \left( {1 - 2z} \right)$$
31. $$Y\left( t \right) = {t^8}{\cos ^4}\left( t \right)$$
32. $$f\left( x \right) = \sqrt {6 - {x^4}} \,\,\,\ln \left( {10x + 3} \right)$$
33. $$A\left( z \right) = \sec \left( {4z} \right)\tan \left( {{z^2}} \right)$$
34. $$h\left( v \right) = \sqrt {5v} + \ln \left( {{v^4}} \right){{\bf{e}}^{6 + 9v}}$$
35. $$\displaystyle f\left( x \right) = \frac{{{{\bf{e}}^{{x^{\,2}} + 8x}}}}{{\sqrt {{x^4} + 7} }}$$
36. $$\displaystyle g\left( x \right) = \frac{{{{\left( {4x + 1} \right)}^3}}}{{{{\left( {{x^2} - x} \right)}^6}}}$$
37. $$\displaystyle g\left( t \right) = \frac{{\csc \left( {1 - t} \right)}}{{1 + {{\bf{e}}^{ - t}}}}$$
38. $$\displaystyle V\left( z \right) = \frac{{{{\sin }^2}\left( z \right)}}{{1 + \cos \left( {{z^2}} \right)}}$$
39. $$U\left( w \right) = \ln \left( {{{\bf{e}}^w}\cos \left( w \right)} \right)$$
40. $$h\left( t \right) = \tan \left( {\left( {5 - {t^2}} \right)\ln \left( t \right)} \right)$$
41. $$\displaystyle z = \ln \left( {\frac{{3 + x}}{{2 - {x^2}}}} \right)$$
42. $$\displaystyle g\left( v \right) = \sqrt {\frac{{{{\bf{e}}^v}}}{{7 + 2v}}}$$
43. $$f\left( x \right) = \sqrt {{x^2} + \sqrt {1 + 4x} }$$
44. $$u = {\left( {6 + \cos \left( {8w} \right)} \right)^5}$$
45. $$h\left( z \right) = {\left( {7z - {z^2} + {{\bf{e}}^{5{z^{\,2}} + z}}} \right)^{ - 4}}$$
46. $$A\left( y \right) = \ln \left( {7{y^3} + {{\sin }^2}\left( y \right)} \right)$$
47. $$g\left( x \right) = {\csc ^6}\left( {8x} \right)$$
48. $$V\left( w \right) = \sqrt{{\cos \left( {9 - {w^2}} \right) + \ln \left( {6w + 5} \right)}}$$
49. $$h\left( t \right) = \sin \left( {{t^3}{{\bf{e}}^{ - 6t}}} \right)$$
50. $$B\left( r \right) = {\left( {{{\bf{e}}^{\sin \left( r \right)}} - \sin \left( {{{\bf{e}}^r}} \right)} \right)^8}$$
51. $$f\left( z \right) = {\cos ^2}\left( {1 + {{\cos }^2}\left( z \right)} \right)$$
52. Find the tangent line to $$f\left( x \right) = {\left( {2 - 4{x^2}} \right)^5}$$ at $$x = 1$$.
53. Find the tangent line to $$f\left( x \right) = {{\bf{e}}^{2x + 4}} - 8\ln \left( {{x^2} - 3} \right)$$ at $$x = - 2$$.
54. Determine where $$A\left( t \right) = {t^3}{\left( {9 - t} \right)^4}$$ is increasing and decreasing.
55. Is $$h\left( x \right) = {\left( {2x + 1} \right)^4}{\left( {2 - x} \right)^5}$$ increasing or decreasing more in the interval $$\left[ { - 2,3} \right]$$?
56. Determine where $$\displaystyle U\left( w \right) = 3\cos \left( {\frac{w}{2}} \right) + w - 3$$ is increasing and decreasing in the interval $$\left[ { - 10,10} \right]$$.
57. If the position of an object is given by $$s\left( t \right) = 4\sin \left( {3t} \right) - 10t + 7$$. Determine where, if anywhere, the object is not moving in the interval $$\left[ {0,4} \right]$$.
58. Determine where $$f\left( x \right) = 6\sin \left( {2x} \right) - 7\cos \left( {2x} \right) - 3$$ is increasing and decreasing in the interval $$\left[ { - 3,2} \right]$$.
59. Determine where $$H\left( w \right) = \left( {{w^2} - 1} \right){{\bf{e}}^{2 - {w^{\,2}}}}$$ is increasing and decreasing.
60. What percentage of $$\left[ { - 3,5} \right]$$ is the function $$g\left( z \right) = {{\bf{e}}^{{z^2} - 8}} + 3{{\bf{e}}^{1 - 2{z^2}}}$$ decreasing?
61. The position of an object is given by $$s\left( t \right) = \ln \left( {2{t^3} - 21{t^2} + 36t + 200} \right)$$. During the first 10 hours of motion (assuming the motion starts at $$t = 0$$) what percentage of the time is the object moving to the right?
62. For the function $$\displaystyle f\left( x \right) = 1 - \frac{x}{2} - \ln \left( {2 + 9x - {x^2}} \right)$$ determine each of the following.
1. The interval on which the function is defined.
2. Where the function is increasing and decreasing.