11X1 T14 06 derivative times function

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  • 1. f x f x dxn

2. f x f x dx n f x n1 c f x f x dx n n 1 3. f x f x dxn f x n1 c f x f x dx nn 1 e.g. i a) Find d dx 1 x 3 4. f x f x dxn f x n1 c f x f x dx nn 1 e.g. i a) Find d dx 1 x 31 x 3 1 x 3 2 3 x 2 1 d1 dx 2 3x 22 1 x3 5. x2 b) Hence find; 1 x 3 dx 6. x2 b) Hence find; 1 x 3 dxx22 3x 2 1 x 3 dx 3 2 1 x 3dx 7. x2 b) Hence find; 1 x 3 dxx22 3x 2 1 x 3 dx 3 2 1 x 3dx2 1 x3 c 3 8. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx 9. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx 1 2 x 2 x 2 dx 2 10. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx 1 2 x 2 x 2 dx 2 2 x 2 2 c3 1 2 2 3 11. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx 1 2 x 2 x 2 dx 2 2 x 2 2 c3 1 2 2 3 2 x 2 2 x 2 c 1 3 12. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx OR x 2 x 2 dx 1 2 x 2 x 2 dx 2 2 x 2 2 c3 1 2 2 3 2 x 2 2 x 2 c 1 3 13. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx OR x 2 x 2 dx u 2 x2 1 2 x 2 x 2 dx 2 2 x 2 2 c3 1 2 2 3 2 x 2 2 x 2 c 1 3 14. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx OR x 2 x 2 dxu 2 x2 1 du 2 x 2 x 2 dx 2x 2 dx du 2 xdx 2 x 2 2 c3 1 2 2 3 2 x 2 2 x 2 c 1 3 15. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx OR x 2 x 2 dxu 2 x2 1 du 2 x 2 x 2 dxu 2x 2 dx du 2 xdx 2 x 2 2 c3 1 2 2 3 2 x 2 2 x 2 c 1 3 16. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx OR x 2 x 2 dxu 2 x2 1 du 2 x 2 x 2 dx1 u 2x 2du dx2 du 2 xdx 2 x 2 2 c3 1 2 2 3 2 x 2 2 x 2 c 1 3 17. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx OR x 2 x 2 dxu 2 x2 1 du 2 x 2 x 2 dx 1 u 2x 2 dudx 2 du 2 xdx 2 x 2 2 c3 1 21 1 2 2 3 u du 2 2 x 2 2 x 2 c 1 3 18. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx OR x 2 x 2 dxu 2 x2 1 du 2 x 2 x 2 dx 1u 2x 2 dudx 2 du 2 xdx 2 x 2 2 c3 1 21 1 2 2 3 u du 2 2 x 2 2 x 2 c 1 1 2 233 u c 2 3 19. x2 b) Hence find; 1 x 3 dx x22 3x 2 1 x 3dx 3 2 1 x 3 dx 2 1 x3 c3 ii x 2 x 2 dx OR x 2 x 2 dx u 2 x2 1du 2 x 2 x 2 dx 1u 2x 2 du dx 2du 2 xdx 2 x 2 2 c3 1 21 1 2 2 3 u du 2 2 x 2 2 x 2 c 1 1 2 233 u c 2 3 2 x 2 2 x 2 c 1 3 20. 1x2 iii dx0x 3 2 3 21. 1x2 iii dx0x 3 2 3 1 1 3x 2 dx 3 0 x 2 3 3 22. 1x2 iii dx0x 3 2 3 1 1 3x 2 dx 3 0 x 2 3 3 1 3 x x 2 dx 1 23 330 23. 1 x2 iii dx0x3 23 1 1 3x 2 dx 3 0 x 2 3 3 1 3 x x 2 dx123 3 301 36 x 2 02 1 24. 1 x2 iii dx0x3 23 1 1 3x 2 dx 3 0 x 2 3 3 1 3 x x 2 dx123 3 301 36 x 2 02 1 1 11 26 1 2 2 1 8 25. 11 x 2x2 iii dx xdx 2OR0x3 2 3 033 1 1 3x 2 dx 3 0 x 2 3 3 1 3 x x 2 dx123 3 301 36 x 2 02 11 11 26 1 2 2 1 8 26. 11 x 2x2u x3 2 iii dx xdx 2OR0x3 2 3 033du 3 x 2 dx 1 1 3x 2 dx 3 0 x 2 3 3 1 3 x x 2 dx123 3 301 36 x 2 02 11 11 26 1 2 2 1 8 27. 11 x 2x2u x3 2 iii dx xdx 2OR0x3 2 3 033du 3 x 2 dxx 0, u 2 1 1 3x 2 dx 3 0 x 2 3 3x 1, u 1 1 3 x x 2 dx123 3 301 36 x 2 02 11 11 26 1 2 2 1 8 28. 1 1 x 2 x2u x3 2 iii dx xdx 2OR0x3 2 30 33 du 3 x 2 dx1 x 0, u 2 1 1 3x 2 1 3 dx u du 3 0 x 2 3 33 2 x 1, u 1 1 3 x x 2 dx123 3 301 36 x 2 02 11 11 26 1 2 2 1 8 29. 1 1 x 2 x2u x3 2 iii dx xdx 2OR0x3 2 30 33 du 3 x 2 dx1 x 0, u 2 1 1 3x 2 1 3 dx u du 3 0 x 2 3 33 2 x 1, u 1 u 2 1 3 x x 2 dx123 31 2 13061 36 x 2 02 11 11 26 1 2 2 1 8 30. 1 1 x 2 x2u x3 2 iii dx xdx 2OR0x3 2 30 33 du 3 x 2 dx1 x 0, u 2 1 1 3x 2 1 3 dx u du 3 0 x 2 3 33 2 x 1, u 1 u 2 1 3 x x 2 dx123 31 2 1306 6 x 2 01 3 2 1 1 1 21 26 1 2 1 11 1 2 6 1 2 28 1 8 31. 1 1 x 2 x2u x3 2 iii dx xdx 2 OR0x3 2 30 33 du 3 x 2 dx1 x 0, u 2 1 1 3x 21 3 dx u du 3 0 x 2 3 3 3 2x 1, u 1 u 2 1 3 x x 2 dx123 3 1 2 130 6 6 x 2 01 3 2 11 1 2 1 2 6 1 2 1 11 1 26 1 2 2 8 1 8Exercise 11H; 1, 3, 5, 7ace etc, 8bdf,9 11*