2. Rules for differentiation:
1. The derivative of a constant function is zero.
example
2. The power rule:
example
[ ] 0=c
dx
d
[ ] 030 =
dx
d
[ ] 1−
= nn
nxx
dx
d
[ ] 3144
12)4(33 xxx
dx
d
== −
Unit 5: Differentiation
5.1 Concepts and Rules of Differentiation
3. 3. The scalar multiple rule:
If f(x) is differentiable at x and c is any real number, then
example
4. The sum rule & difference rule:
example
[ ] [ ])()( xf
dx
d
cxcf
dx
d
= [ ] [ ] 3344
12)4(333 xxx
dx
d
x
dx
d
===
[ ] [ ] [ ])()()()( xg
dx
d
xf
dx
d
xgxf
dx
d
±=±
[ ] 12)5()()(5 22
−=+−=+− x
dx
d
x
dx
d
x
dx
d
xx
dx
d
Unit 5: Differentiation
5.1 Concepts and Rules of Differentiation
4. Rules for differentiation:
5. The Product Rule:
Example: Differentiate
Solution:
[ ] )(')()().(')().( xgxfxgxfxgxf
dx
d
+=
)1)(13( 2
xxy +−=
[ ]
169
1366
)1)(13()6)(1(
)1()13()13()1()1)(13(
2
22
2
222
−+=
−++=
−++=
+−+−+=+−
xx
xxx
xxx
x
dx
d
xx
dx
d
xxx
dx
dy
Unit 5: Differentiation
5.1 Concepts and Rules of Differentiation
5. 6. The Quotient Rule: If
then
Example:
Differentiate with respect to x .
,0)(,
)(
)(
)( ≠= xh
xh
xg
xf
2
)]([
)()(')()('
)('
xh
xgxhxhxg
xf
×−×
=
23
12
)(
+
−
=
x
x
xf
Unit 5: Differentiation
5.1 Concepts and Rules of Differentiation
6. 7. The Chain Rule:
If is the composite function in which the
inside function g and the outside function f are
differentiable, then
Example:
Differentiate with respect to x .
))(( xgf
Unit 5: Differentiation
5.1 Concepts and Rules of Differentiation
)(')).(('))](([ xgxgfxgf
dx
d
=
( )22
1−+= xxy
7. The derivative of an Exponential Function:
Example: Determine the derivative of
Solution:
Unit 5: Differentiation
5.1 Diff. of Exp. & Log. Function
( ) xx
ee
dx
d
=
x
xexf 3)( =
)3()(3)3()(' x
dx
d
ee
dx
d
xxe
dx
d
xf xxx
+==
)1(333 +=+= xeexe xxx
8. The derivative of a Natural Logarithm Function:
where x > 0
And by the chain rule:
where f (x) > 0
Unit 5: Differentiation
5.1 Diff. of Exp. & Log. Function
x
x
dx
d 1
)(ln =
)(
)('
)]([ln
xf
xf
xf
dx
d
=
9. Example:
Find the derivative of
Solution:
Unit 5: Differentiation
5.1 Diff. of Exp. & Log. Function
)52ln()( 2
−+= xxxf
52
22
)52(
)52(
)(' 22
2
−+
+
=
−+
−+
=
xx
x
xx
xx
dx
d
xf
10. Marginal functions in economics
Example: The total cost, C(x) associated with the producing
and marketing x units of MP4 player is given by
Find :
(a)The actual cost incurred for producing and marketing the
11th
MP4 player.
(b) Total cost when the output is 5 units.
(c) Average cost for an output of 10 units
(d) Marginal cost the output is 4 units.
Unit 5: Differentiation
5.2 Business Application
150201.05.0)( 23
+−−= xxxxC
11. Marginal functions in economics
Example: Suppose the relationship between the unit price p in
RM and the quantity demanded x of the laptop model S is
given by the equation
Find :
(a) the revenue function R(x)
(b) the marginal revenue foundation R’(x)
(c) Compute R’(200) .
Unit 5: Differentiation
5.2 Business Application
30002.0)( +−= xxp
12. Marginal functions in economics
Example: For the above example , the revenue is function is
. Let the total cost of producing
x units of laptop model S be
Find :
(a) the profit function P(x)
(b) the marginal profit function P’(x)
(c) Compute P’(100)
Unit 5: Differentiation
5.2 Business Application
xxx 30002.0)( 2
+−=
300000200)( += xxC
13. Unit 5 Integration
5.3 Integration as Anti-differentiation
Definition:
A function F is called an anti-derivative or an indefinite integral
of a function f(x) if the derivative F’(x) = f(x). We write
if
Example:
Since the derivative of x2
= 2x, or
therefore, x2
is an anti-derivative of 2x, or
∫= dxxfxF )()( )()(' xfxF =
xx
dx
d
2)( 2
=
∫ = 2
2 xxdx
14. Unit 5 Integration
5.3 Integration as Anti-differentiation
x2
is not the only anti-derivative of 2x,
The derivative of x2
+ 5 is also 2x,
therefore x2
+ 5 is also anti-derivative of 2x.
Constant term is the only difference
In general,
it is always necessary to include the constant C
when writing a general anti-derivative:
x2
+ C
15. Unit 5 Integration
5.3 Integration as Anti-differentiation
Integration:
If then, integrating f(x) produces
the anti-derivative F(x) + C.
We write:
[ ] )()( xfxF
dx
d
=
∫ += CxFdxxf )()(
16. Unit 5 Integration
5.3 Integration as Anti-differentiation
Basic Integration Rules:
Rule 1: The indefinite integral of a constant
Example:
∫ += Ckxkdx
∫ += Cxdx 99
17. Unit 5 Integration
5.3 Integration as Anti-differentiation
Basic Integration Rules:
Rule 2: The power rule
Example:
)( 1
1
1 1
−≠+
+
=∫
+
nCx
n
dxx nn
∫ +=+
+
= +
CxCxdxx 5144
5
1
14
1
20. Unit 5 Integration
5.3 Integration as Anti-differentiation
Basic Integration Rules:
Rule 5: The indefinite integral of the exponential
function
(k = constant)
Example:
∫ += Ce
k
dxe kxkx 1
∫ += −−
Cedxe xx 8585
5
1
21. Unit 5 Integration
5.3 Integration as Anti-differentiation
Basic Integration Rules:
Rule 6: The indefinite integral of
Example:
x
xxf
11
== −
)(
∫ ∫ ≠+==−
0
11
xCxdx
x
dxx ln
∫ ∫ ≠+==−
)(ln 05
1
55 1
xCxdx
x
dxx
22. Unit 5 Integration
5.3 Definite Integrals
Theorem:
Let f be continuous on [a, b]. If G is any anti-derivative
for f on [a, b], then
Example: Evaluate .
Solution: Since is an antiderivative of f(x) = x, thus
∫ −=
b
a
aGbGdxxf )()()(
∫
2
1
dxx
2
)(
2
x
xF =
2
3
)14(
2
1
2
2
1
2
2
1
=−==∫
x
dxx
23. Unit 5 Integration
5.3 Definite Integrals
Evaluating the definite Integral:
Let f and g be continuous function, then
( c = constant)
[ ]
( )cbadxxfdxxfdxxf
dxxgdxxfdxxgxf
dxxfcdxxcf
dxxfdxxf
dxxf
c
b
b
a
c
a
b
a
b
a
b
a
b
a
b
a
b
a
a
b
a
a
<<+=
±=±
=
−=
=
∫∫∫
∫∫∫
∫ ∫
∫ ∫
∫
)()()(.
)()()()(.
)()(.
)()(.
)(.
5
4
3
2
01