Feb 13 lecture
(x
p
)
′
= px
p-1
(cf)
′
= cf
′
(f+g)
′
= f
′
+g
′
(fg)
′
= f
′
g + f g
′
(1/g)
′
= - g
′
/ g
2
(f/g)
′
= (f
′
g - f g
′
) / g
2
(f(g))
′
= f
′
(g) g
′
(e
x
)
′
= e
x
(e
g
)
′
= e
g
g
′
Practice Problem.
Find the derivative of
2
x
x e
fg = f
′
g + f g
′
(e
g
)
′
= e
g
g
′
2
2
1/ 2
3/ 2
(
)'
2
1
2
2
2
x
x
x
x
x e
x e
x
e
x
x
x
e
−
=
⋅
⎛
⎞
+
⋅
⎜
⎟
⎝
⎠
⎛
⎞
=
+
⎜
⎟
⎝
⎠
Bases other than e
(
)
(
)
ln3
ln3
ln3
( )
3
Find
'( )
Recall
(
)'
'
3
3
'
(ln3)
(ln3)3
x
g
g
x
x
x
x
x
x
f x
f
x
e
e g
e
e
e
=
=
=
=
=
=
Newton’s law of cooling
An object of temperature
in a
room of temperature
(
)
(
( )
)
(
(0)
)
difference is reduced by 1/2 after time ln 2/
kt
T
R
dT
k T
R
dt
T t
R
T
R e
k
−
= −
−
−
=
−
CSI-Ithaca
0.0676
A dead body in a 68
room
( )
68
30.6
T t
e
−
=
+
D
Coroner’s rule of thumb:
cools 2 degrees in first hour
and 1 degree per hour after that

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