Page 105 - Calculus Workbook For Dummies

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Chapter 6: Rules, Rules, Rules: The Differentiation Handbook

F For y = sinx + cosx, find the 1st through 6th derivatives.
y = l cos x - sin x
ll
y = - sin x - cos x
lll
y = - cos x + sin x
4
( )
y = sin x + cos x
( )
5
y = cos x - sin x
( )
6
y = - sin x - cos x
Notice that the 4th derivative equals the original function; that the 5th derivative equals the
1st, and so on. This cycle of four functions repeats ad infinitum.
2
G For y = cosx , find the 1st, 2nd, and 3rd derivatives.
y = cosx 2
l
y =- 2 x sinx 2 ^ chain ruleh
2 l
x l
2
y = - 2 h _ sinx i + - 2 x _ h sinx i _ product rulei
^
ll ^
2
2
= - 2 sinx - 2 x cos x i 2 x ^ chain ruleh
_
2
= - 2 sinx - 4 x 2 cosx 2
2 l
2 l
2
2
2
lll
y = - 2 cos x i 2 x - _ 9 4 x i _ cosx i + _ 4 x _i cosx i C
_
2
2
2
2
= - 4 x cosx - ; 8 x cosx + 4 a sin x i 2 xkE
x -
_
2
2
= - 4 x cosx - 8 x cosx + 8 x 3 sinx 2
2
= 8 x 3 sinx - 12 x cosx 2