Calculus I
Formula Sheet
Chapter 3
Section 3.1
1. Definition of the derivative of a function:
( )
0
( ) ()
lim
x
fx x fx
fx x
∆→
+∆ −
′=∆
2. Alternative form of the derivative at
:xc=
( )
() ()
lim
xc
fx fc
fc xc
→
−
′=−
3. Differentiability
⇒
Continuity
Section 3.2
4.
[ ]
0
dc
dx =
5.
[ ]
1
dx
dx =
6.
1nn
dx nx
dx
−
=
7.
( ) ( )
dcf x cf x
dx ′
=
8.
[ ]
() () () ()
dfxgx fxgx
dx ′′
+=+
9.
[ ]
() () () ()
dfxgx fxgx
dx ′′
−=−
10.
[ ]
sin cos
dxx
dx =
11.
[ ]
cos sin
dxx
dx = −
12.
xx
dee
dx
=
13. Free Fall:
200
1
() 2
s t gt v t s=− ++
Feet: -32 Meters: -9.8
14. Velocity:
( )
()vt st
′
=
15. Find equation of tangent line to curve at
:xc=
a. Equation of Line: need point and slope
b. Point:
( )
( )
,cfc
Slope:
()m fc
′
=
c. Point-Slope form:
( )
11
y y mx x−= −
16. Derivative: a) Slope of tangent line
b) Instantaneous rate of change
Section 3.3
17.
[ ]
( ) ( ) () ( )
() ()
df x gx f xg x g x f x
dx ′′
⋅= +
18.
( ) ( ) ( ) ( )
[ ]
2
()
() ()
gx f x f xg x
d fx
dx g x gx
′′
−
=
19.
[ ]
2
tan sec
dxx
dx =
20.
[ ]
sec sec tan
dx xx
dx =
21.
[ ]
2
cot csc
dxx
dx = −
22.
[ ]
csc csc cot
dx xx
dx = −
23. Horizontal Tangent Line:
() 0m fx
′
= =
Section 3.4
24. Chain Rule:
( ), ( ), dy dy du
y yu u ux dx du dx
= = = ⋅
25.
[ ]
1
ln
dx
dx x
=
26. Definition of exponential function to base a:
( )
lnax
x
ae=
27. Definition of logarithmic function to base a:
1
log ln
ln
a
xx
a
=
28.
( )
ln
xx
da aa
dx
=
29.
[ ]
1
log (ln )
a
dx
dx a x
=
Section 3.5
30. Implicit Differentiation:
a. Differentiate both sides with respect to x.
b. Isolate
dy
dx
on the left side.
31. Logarithmic Differentiation
a. Take a ln of both sides
b. Use properties of ln to expand
c. Differentiate implicitly
