Section 3.9 : Chain Rule

For problems 1 – 51 differentiate the given function.

1. \(g\left( x \right) = {\left( {3 - 8x} \right)^{11}}\)
2. \(g\left( z \right) = \sqrt[7]{{9{z^3}}}\)
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. \(A\left( t \right) = \cos \left( t \right) - \sqrt[6]{{1 - \sin \left( t \right)}}\)
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. \(h\left( x \right) = \sqrt[3]{{{x^2} - 5x + 1}} + {\left( {9x + 4} \right)^{ - 7}}\)
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. \(V\left( w \right) = \sqrt[4]{{\cos \left( {9 - {w^2}} \right) + \ln \left( {6w + 5} \right)}}\)
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.