writeup.ms (1505B)
1 .2C 2 3 .NH 4 Differentiating standard equations 5 .EQ L 6 dy over dx = n x sup n-1 7 .EN 8 .NH 2 9 Example 10 .LP 11 Define the equation 12 .EQ L 13 y = 5 x sup 2 + 19 x + 8 14 .EN 15 Use the formula 16 .EQ L 17 dy over dx = 10 x + 19 18 .EN 19 Its very simple 20 .NH 21 Differentiating trig equations 22 .LP 23 From the chain rule, one can find the following: 24 .EQ L 25 sin(kx) -> k cos(kx) 26 .EN 27 .EQ L 28 cos(kx) -> -k sin(kx) 29 .EN 30 .EQ L 31 tan(kx) -> k sec sup 2 (kx) 32 .EN 33 .EQ L 34 sec(kx) -> k sec(kx) tan(kx) 35 .EN 36 .EQ L 37 cot(kx) -> -k cosec sup 2 (kx) 38 .EN 39 .EQ L 40 cosec(kx) -> -k cosec(kx) cot(kx) 41 .EN 42 .NH 43 Chain rule 44 .EQ L 45 dy over dx = dy over dt times dt over dx 46 .EN 47 .NH 2 48 Example 49 .LP 50 Define the function 51 .EQ L 52 y =sin sup 2 (9x) 53 .EN 54 Re-write y in terms of t 55 .EQ L 56 Y =sin sup 2 (t) 57 .EN 58 Define t 59 .EQ L 60 t = 9x 61 .EN 62 Differentiate y with respect to t 63 .EQ L 64 dy over dt = 2cos(t) 65 .EN 66 Differentiate y with respect to x 67 .EQ L 68 dt over dx = 9 69 .EN 70 Times the two together 71 .EQ L 72 dy over dx = 2cos(t) times 9 73 .EN 74 Substute the original t back in 75 .EQ L 76 dy over dx = 18cos(9x) 77 .EN 78 79 .NH 80 Product rule 81 82 .EQ L 83 "when " y = u v 84 .EN 85 .EQ L 86 dy over dx = ( v prime times u ) + ( v times u prime ) 87 .EN 88 .NH 2 89 Example 90 91 .LP 92 Define the equation 93 .EQ L 94 y = sin(x) cos(x) 95 .EN 96 Define u and v 97 .EQ L 98 u = sin(x) 99 .EN 100 .EQ L 101 v = cos(x) 102 .EN 103 Differentiate indiviually 104 .EQ L 105 u prime = cos(x) 106 .EN 107 .EQ L 108 v prime = -sin(x) 109 .EN 110 Put into the formula 111 .EQ L 112 ( v prime times u ) + ( v times u prime ) = -sin(x)sin(x) + cos(x)cos(x) 113 .EN 114 Simplify 115 .EQ L 116 dy over dx = -sin sup 2 (x) +cos sup 2 (x) 117 .EN 118 .EQ L 119 dy over dx = cos(2x) 120 .EN 121