Paul's Online Notes
Home / Calculus I / Derivatives / Chain Rule
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

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. $$y = \sqrt{{1 - 8z}}$$ 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. $$f\left( x \right) = {\left( {\sqrt{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)^{ - 1}}$$ 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