Page 100 - Calculus Workbook For Dummies

84 Part III: Differentiation

4. Complete the problem as shown in Step 2.
2 x l 2 x lnx = l x 2 x 2 x 1
h
h
_ x e i ^ lnx + _ x e ^ i h _ 2 xe + x e ^ i lnx + x e $ x
2
= 2 xe x lnx + x e x lnx + xe x or
2
= x e x lnx + 2 xe x lnx + xe x or
x
= xe x lnx + 2 lnx + h
1 or
^
2
x +
= e x lnx x + 2 i xe x
_
You say “pa-ja-mas,” I say “po-jah-mas.”
d x 3 3 x 2 cosx + x 3 sinx
m =
dx cosx cos x
2
3 l 3 cosx l
h
d x 3 = _ x i ^ cosx - _ x ^ i h
dx cosx ^ cosxh 2
3
3 x 2 cosx - x - sinxh
^
=
2
cos x
3 x 2 cosx + x 3 sinx
= 2
cos x
d
cosx
n dx e x = - sinx - x cosx
e
x l
x
h
_ h
d cosx ^ cosx l _ e i - ^ cosx e i
dx e x = x 2
_ e i
- e x sinx - e x cosx
= 2 x
e
- sinx - cosx
= x
e
d 3 x + 3 6 x arctanx - 3
2
o =
dx arctanx arctan x
2
d 1
A handy-dandy rule: arctanx = 2 .
dx 1 + x
2
2
3 ^
h
h
3 ^
d 3 x + 3 _ 3 x + i l arctanx - _ 3 x + i arctanx l
2
dx arctanx = ^ arctanxh 2
1
6 x arctanx - _ 3 x + i n
2
3 d
2
x + 1
=
arctan x
2
6 x arctanx - 3
=
arctan x
2
To remember the derivatives of the inverse trig functions, notice that the derivative of each co-
function (arccosine, arccotangent, and arccosecant) is the negative of its corresponding function.
So, you really only need to memorize the derivatives of arcsin, arctan, and arcsec. These three
have a 1 in the numerator. The two that contain the letter “s,” arcsin and arcsec, contain a
square root in the denominator. Arctan has no “s” and no square root.