10 ) ( , 5 ) ( x g x f ) ( ' , ) ( ' : x g x f Find 0 c dx d Sec 3.3: Differentiation Rules Example : Constant function
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10)(,5)( xgxf
)(', )(' : xgxfFind
0cdx
d
Sec 3.3: Differentiation Rules
Example:
Constant function
52 )(,)(,)( xxhxxgxxf
)(', )(' : xgxfFind
1 nn nxxdx
d
Sec 3.3: Differentiation Rules
Example:
The power Rule
)(', )(' : xgxfFind
1 nn nxxdx
d
Sec 3.3: Differentiation Rules
Example:
The power Rule
xxgx
xf )(,1
)(
)(' : xfFind
Example:
xxxf )(
)(')( xcfxcfdx
d
43)( xxf
Sec 3.3: Differentiation Rules
)(', )(' : xgxfFind
Example:
The constant multiple
)(')(')()( xgxfxgxfdx
d
832096)( 234 xxxxxf
)(')(')()( xgxfxgxfdx
d
Sec 3.3: Differentiation Rules
)(', )(' : xgxfFind
Example:
The sum and differnce Rule
Sec 3.3: Differentiation Rules
)(' : xfFind
Example:
Derivative of exponential
xx eedx
d)(
2)( xexf x
Definition of the number e
11
lim0
h
eh
h
Sec 3.3: Differentiation Rules
Derivative of exponential
)(ln)( aaadx
d xx
0a
)(' : xfFind
Example:
xxx xhxgxf )(,5)(,2)(
Sec 3.3: Differentiation Rules
TERM-121 Exam-2
Sec 3.3: Differentiation Rules
TERM-121 Exam-2
Sec 3.3: Differentiation Rules
TERM-121 Exam-2
Sec 3.3: Differentiation Rules
)(' : xfFind
)(')()()(')()( xgxfxgxfxgxfdx
d
Example:
Derivative Product Rule
)(1
)( 2 xexx
xf
Sec 3.3: Differentiation Rules
)(' : tyFind
2)]([
)(')()()('
)(
)(
xg
xgxfxgxf
xg
xf
dx
d
Example:
Derivative Quotient Rule
1
1)(
3
2
t
tty
)(' : xyFind
Example:
4
2 )2)(1()(
x
xxxxy
Sec 3.3: Differentiation Rules
)(' xf
Second- and Higher-Order Derivatives
)5(,'',' : yyyFind
Example:
2042)( 25 xxxy
)('' xf
)(''' xf
)()( xf n
First derivative
second derivative
3ed derivative
n-th derivative
)(' xfdx
d
)('' xfdx
d
n
n
dx
fd
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