1
Formulas
Remember, mathematics is all about being lazy. Using the limit deﬁnition of the derivative will get very
diﬃcult if we have to do it every time. Fortunately, we have methods for computing common derivatives
like we have methods for computing common limits.
d
dx
(
c
)
Prove this.
d
dx
(
x
) = 1
Prove this.
d
dx
(
x
n
) =
nx
n

1
Prove this using the binomial theorem.
1.1
Derivative laws (like limit laws)
•
d
dx
(
cf
(
x
))
•
d
dx
(
f
(
x
) +
g
(
x
))
•
d
dx
(
f
(
x
)

g
(
x
))
Use scalar multiplication + addition.
Example 1.1.
Find
f
0
(
x
) if
f
(
x
) =
x
8
+ 7
x
5

20
x
3
+ 17
x
2

1
Example 1.2.
Find the points of
y
=
x
4

6
x
2
+ 4 where the tangent line is horizontal.
1.2
Product and Quotient rules
•
Product Rule
Pull out and
f
(
x
+
h
) and a
g
(
x
).
•
Quotient Rule
Add and subtract
f
(
x
)
g
(
x
)
Example 1.3.
Find
f
0
(
x
) if
f
(
x
) = (6
x
3
)(7
x
4
)
Example 1.4.
Find
d
dx
±
x
2
+
x

2
x
3
+6
²
1.3
Power Functions
Prove
d
dx
(
x

n
) =

n

n

1
using quotient rule.
Example 1.5.
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 Spring '08
 varies
 Trigonometry, Derivative, Formulas, Sin, lim

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