Page 99 - Calculus Workbook For Dummies

83
Chapter 6: Rules, Rules, Rules: The Differentiation Handbook

i d _ x 3 cosx = 3 x 2 cosx - x 3 sinx
i
dx
d
Remember that cosx = - sinx. For a great mnemonic to remember the derivatives of trig
dx
functions, check out Chapter 15.
d 3 3 l 3 cosx l
h
dx _ x cosx = _i x i ^ cosx + _ x ^ i h
3
= 3 x 2 cosx + x - sinxh
^
= 3 x 2 cosx - x 3 sinx
j d ^ sin tanx = cos tanx + sec tanx
x
x
x
h
dx
d 2
A helpful rule: tanx = sec x.
dx
d ^ sin tanx = ^h sinx l ^ tanx + ^h sinx ^h tanx l
x
h
h
dx
2
x
x
= cos tanx + sin sec x
x
x
= cos tanx + sec tanx
d
k dx 5 x 3 lnx = 5 x 3 lnx + 1h
2
^
d 1
Another helpful rule: lnx = . As for the 5, you can deal with it in two ways. First, you can
dx x
ignore it temporarily (because coefficients have no effect on differentiation), and put it back
when you’re done differentiating. If you do it this way, don’t forget that the “5” multiplies the
entire derivative, not just the first term. The second way is probably easier and better: just
make the “5” part of the first function. To wit:
d 5 x 3 lnx = _i 5 x i lnx + _ 5 x ^ i lnx l
3 l
3
dx _ ^ h h
3
= 15 x 2 lnx + 5 x $ 1
x
= 15 x 2 lnx + 5 x 2 or
2
= 5 x 3 ^ lnx + 1h ; take your pick.
*l d x e x lnx = e x lnx x + 2 i xe x
2
2
x +
_
dx
This is a challenge problem because, as you’ve probably noticed, there are three functions
in this product instead of two. But it’s a piece o’ cake. Just make it two functions: either
2 x 2 x
_ x e ^ i lnxh or _ x _i e lnxi. Take your pick.
d x x x x
A handy rule: e = e (Note that e and its multiples (like 4e ) are the only functions that are
dx
their own derivatives.)
1. Rewrite this “triple function” as the product of two functions.
d 2 x
= _ x e ^ i lnxh
dx
2. Apply the product rule.
d 2 x 2 x l 2 x lnx l
h
h
dx _ x e ^ i lnx = _ x e i ^ lnx + _ x e ^ i h
x l
2
3. Apply the product rule separately to x e i , then substitute the answer back where it
_
belongs.
2 x l 2 l x 2 x l
_ x e i = _ x i _ e i + _ x _i e i
x
2
= 2 xe + x e x