CHAPTER 3. RULES FOR DERIVATIVES
Example 1. Find
d
(3x + ex )(x2
dx
90
4ex ).
Solution. Identify f and g:
(3x + ex ) (x2 4ex )
| {z } | {z }
g
f
Calculate f 0 and g 0 :
f 0 = 3 + ex
g 0 = 2x 4ex
Put it all together using the product rule:
f 0 · g + f · g 0 = (3 + ex )(x2
4ex ) + (3x + ex )(2x
4ex ).
CHAPTER 3. RULES FOR DERIVATIVES
d
Example 2. Find
(x + e3x
dx
Solution. Identify f and g:
1
)
✓
◆
1
+x .
x
◆
1
)
+x
} x
| {z }
3x 1
(x + e
| {z
f
Calculate f 0 and g 0
91
✓
g
f 0 = 1 + e3x 1 · 3
1
g0 =
+1
x2
Put it all together using the product rule:
✓
◆
✓
◆
1
1
0
0
3x 1
3x 1
f · g + f · g = (1 + 3e
)·
+ x + (x + e
)·
+1
x
x2
CHAPTER 3. RULES FOR DERIVATIVES
Example 3. Find
92
d 2x + 1
.
dx ex + x
Solution. Identify f and g
2x + 1
ex + x
f
g
Calculate f 0 and g 0
f0 = 2
g 0 = ex + 1
Put it all together using the quotient rule
f0 · g
g2
f · g0
=
2(ex + x)
(2x + 1)(ex + 1)
(ex + x)2
CHAPTER 3. RULES FOR DERIVATIVES
93
p
t+ t
Example 4. Find the derivative of 2
.
t + ln(t)
Solution. Identify f and g
p
t+ t
t2 + ln(t)
f
g
Calculate f 0 and g 0
1
f0 = 1 + p
2 t
1
g 0 = 2t +
t
Put it all together using the quotient rule
f0 · g
g2
f · g0
=
(1 +
2
1
p
)(t2 + ln(t))
t
(t +
(t2 + ln(t))2
p
t)(2t +
1
t
)