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### Section 3-9 : Chain Rule

For problems 1 – 27 differentiate the given function.

1. $$f\left( x \right) = {\left( {6{x^2} + 7x} \right)^4}$$ Solution
2. $$g\left( t \right) = {\left( {4{t^2} - 3t + 2} \right)^{ - 2}}$$ Solution
3. $$y = \sqrt[3]{{1 - 8z}}$$ Solution
4. $$R\left( w \right) = \csc \left( {7w} \right)$$ Solution
5. $$G\left( x \right) = 2\sin \left( {3x + \tan \left( x \right)} \right)$$ Solution
6. $$h\left( u \right) = \tan \left( {4 + 10u} \right)$$ Solution
7. $$f\left( t \right) = 5 + {{\bf{e}}^{4t + {t^{\,7}}}}$$ Solution
8. $$g\left( x \right) = {{\bf{e}}^{1 - \cos \left( x \right)}}$$ Solution
9. $$H\left( z \right) = {2^{1 - 6z}}$$ Solution
10. $$u\left( t \right) = {\tan ^{ - 1}}\left( {3t - 1} \right)$$ Solution
11. $$F\left( y \right) = \ln \left( {1 - 5{y^2} + {y^3}} \right)$$ Solution
12. $$V\left( x \right) = \ln \left( {\sin \left( x \right) - \cot \left( x \right)} \right)$$ Solution
13. $$h\left( z \right) = \sin \left( {{z^6}} \right) + {\sin ^6}\left( z \right)$$ Solution
14. $$S\left( w \right) = \sqrt {7w} + {{\bf{e}}^{ - w}}$$ Solution
15. $$g\left( z \right) = 3{z^7} - \sin \left( {{z^2} + 6} \right)$$ Solution
16. $$f\left( x \right) = \ln \left( {\sin \left( x \right)} \right) - {\left( {{x^4} - 3x} \right)^{10}}$$ Solution
17. $$h\left( t \right) = {t^6}\,\sqrt {5{t^2} - t}$$ Solution
18. $$q\left( t \right) = {t^2}\ln \left( {{t^5}} \right)$$ Solution
19. $$g\left( w \right) = \cos \left( {3w} \right)\sec \left( {1 - w} \right)$$ Solution
20. $$\displaystyle y = \frac{{\sin \left( {3t} \right)}}{{1 + {t^2}}}$$ Solution
21. $$\displaystyle K\left( x \right) = \frac{{1 + {{\bf{e}}^{ - 2x}}}}{{x + \tan \left( {12x} \right)}}$$ Solution
22. $$f\left( x \right) = \cos \left( {{x^2}{{\bf{e}}^x}} \right)$$ Solution
23. $$z = \sqrt {5x + \tan \left( {4x} \right)}$$ Solution
24. $$f\left( t \right) = {\left( {{{\bf{e}}^{ - 6t}} + \sin \left( {2 - t} \right)} \right)^3}$$ Solution
25. $$g\left( x \right) = {\left( {\ln \left( {{x^2} + 1} \right) - {{\tan }^{ - 1}}\left( {6x} \right)} \right)^{10}}$$ Solution
26. $$h\left( z \right) = {\tan ^4}\left( {{z^2} + 1} \right)$$ Solution
27. $$f\left( x \right) = {\left( {\sqrt[3]{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)^{ - 1}}$$ Solution
28. Find the tangent line to $$f\left( x \right) = 4\sqrt {2x} - 6{{\bf{e}}^{2 - x}}$$ at $$x = 2$$. Solution
29. Determine where $$V\left( z \right) = {z^4}{\left( {2z - 8} \right)^3}$$ is increasing and decreasing. Solution
30. The position of an object is given by $$s\left( t \right) = \sin \left( {3t} \right) - 2t + 4$$. Determine where in the interval $$\left[ {0,3} \right]$$ the object is moving to the right and moving to the left. Solution
31. Determine where $$A\left( t \right) = {t^2}{{\bf{e}}^{5 - t}}$$ is increasing and decreasing. Solution
32. Determine where in the interval $$\left[ { - 1,20} \right]$$ the function $$f\left( x \right) = \ln \left( {{x^4} + 20{x^3} + 100} \right)$$ is increasing and decreasing. Solution